矩阵函数的谱自适应多项式逼近及其在图形信号处理中的应用

摘要

我们提出并研究了两种新的近似方法$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$对于大型稀疏厄米矩阵$\mathbf{<trans\; data-src="A">A类</trans>}$这种形式的计算在许多信号处理和机器学习任务中发挥着重要作用。这两种方法的主要思想是首先估计$\mathbf{<trans\; data-src="A">A类</trans>}$，然后找到更接近函数的固定阶多项式（f）具有较高特征值密度的谱区域。与Lanczos方法和截断Chebyshev展开等最先进的方法相比，所提出的方法倾向于提供更精确的近似值$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$低多项式阶，对于矩阵$\mathbf{<trans\; data-src="A">A类</trans>}$具有大量不同的内部特征值和较小的谱宽。我们还探索了这些技术在以下方面的应用：（i）快速估计局部化图谱滤波器字典原子的范数，以及（ii）快速过滤时间顶点信号。

1.简介

高效计算$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$，是大型稀疏埃尔米特矩阵乘以向量的函数，是许多信号处理、机器学习、应用数学和计算机科学任务的重要组成部分。应用示例包括基于图形的半监督学习方法[1,2,三]; 图形信号处理中的图形谱滤波[4]; 卷积神经网络/深度学习[5,6]; 群集[7,8]; 大矩阵谱密度的逼近[9]; 矩阵数值秩的估计[10,11]; 近似谱和，例如矩阵的对数行列式[12]或矩阵求逆的轨迹，用于物理、生物学、信息论和其他学科[13]; 求解半定规划[14]; 模拟随机行走[15]（第八章）；解常微分方程和偏微分方程[16,17,18].

参考文献[19]（第13章）[20,21,22]调查解决这个研究得很好的高效计算问题的不同方法

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\mathbf{<trans\; data-src="V">五</trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="\Lambda ">\Lambda </trans>}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="V">五</trans>}}^{<trans\; data-src="\top ">\top </trans>}\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>\end{array}$$

(1)

其中的列$\mathbf{<trans\; data-src="V">五</trans>}$是厄米矩阵的特征向量$\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$；$\mathbf{<trans\; data-src="\Lambda ">\Lambda </trans>}$是一个对角矩阵，其对角元素是$\mathbf{<trans\; data-src="A">A类</trans>}$，我们用${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}}$; 和$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="\Lambda ">\Lambda </trans>}<trans\; data-src=")">)</trans>$是一个对角矩阵，其k个对角线条目为$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans>$对于大型矩阵，显式计算$\mathbf{<trans\; data-src="A">A类</trans>}$为了近似(1). 相反，最常见的技术，所有这些都避免了$\mathbf{<trans\; data-src="A">A类</trans>}$，包括（i）截断正交多项式展开式，包括切比雪夫[23,24,25]和雅各比；（ii）合理近似[21]（第3.4节）；（iii）Krylov子空间方法，如Lanczos方法[23,26,27,28,29]; 和（iv）求积/轮廓积分方法[19]（第13.3节）。

我们在这项工作中的重点是多项式近似方法。让${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="K">K（K）</trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$成为学位K（K）函数的多项式逼近（f）在已知间隔上$<trans\; data-src="[">[</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="]">]</trans>$包含的所有特征值$\mathbf{<trans\; data-src="A">A类</trans>}$。然后是近似值${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$可以通过特定类型多项式的三项递归进行递归计算（参见第3节更多详细信息），或通过嵌套的乘法迭代[30]（第9.2.4节），出租${\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}\mathbf{<trans\; data-src="b">b条</trans>}$，然后迭代

$$\begin{array}{c}\hfill {\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="l">我</trans>}}\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="+">+</trans>\mathbf{<trans\; data-src="A">A类</trans>}{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=".">。</trans>\end{array}$$

(2)

这两种方法的计算成本都取决于稀疏矩阵的乘法$\mathbf{<trans\; data-src="A">A类</trans>}$通过K（K）不同的矢量。近似误差的范围是

$$\begin{array}{cc}\hfill <trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="\Vert ">\Vert </trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}& <trans\; data-src="=">=</trans>\underset{<trans\; data-src="\ell ">Ş</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="max">最大值</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & <trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="]">]</trans>}{<trans\; data-src="sup">啜饮</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans><trans\; data-src=".">。</trans>\hfill \end{array}$$

（4）

例如，如果，${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$是一个学位K（K）解析函数的截断切比雪夫级数逼近（f），（）中的上界在几何上收敛到0，如下所示K（K）增加，速度为$\mathcal{<trans\; data-src="O">O（运行）</trans>}\left({\mathrm{<trans\; data-src="\rho ">\rho </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}\right)$，其中$\mathrm{<trans\; data-src="\rho ">\rho </trans>}$是开放伯恩斯坦椭圆的半径（f）是解析的和有界的（参见例如[31]（定理5.16）[32]（定理8.2））。

除了计算效率和收敛性保证外，多项式近似方法的第三个优点是可以在分布式环境中实现[33]. 第四个优点是我的第个元素${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$只取决于$\mathbf{<trans\; data-src="b">b条</trans>}$在内部K（K）啤酒花我关于关联图$\mathbf{<trans\; data-src="A">A类</trans>}$（例如，如果$\mathbf{<trans\; data-src="A">A类</trans>}$是一个图拉普拉斯矩阵，是$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>$的第个元素$\mathbf{<trans\; data-src="A">A类</trans>}$，其中$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="j">j个</trans>}$，对应于连接顶点的边我和j个（如图所示）。这种定位特性在许多基于图形的数据分析应用中都很重要，例如图谱滤波[34]和深度学习[5]. 最后，与其他包含有关$\mathbf{<trans\; data-src="b">b条</trans>}$选择近似多项式（例如[35]考虑向量$\mathbf{<trans\; data-src="b">b条</trans>}$从具有已知协方差矩阵的零米分布），我们提出的方法得到的多项式近似不依赖于$\mathbf{<trans\; data-src="b">b条</trans>}$或有关的任何信息$\mathbf{<trans\; data-src="b">b条</trans>}$因此，在计算$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$对许多不同的矢量重复$\mathbf{<trans\; data-src="b">b条</trans>}$用同样的（f）和$\mathbf{<trans\; data-src="A">A类</trans>}$，多项式系数只需计算一次。

而经典的截断正交多项式展开方法（例如Chebyshev、Legendre、Jacobi）旨在近似函数（f）在整个间隔期间$<trans\; data-src="[">[</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="]">]</trans>$，它只是特征值处的多项式近似误差$\mathbf{<trans\; data-src="A">A类</trans>}$影响中的总体错误(三). 有了完整的特征值集的知识，我们可以做得更好，例如，通过拟合一个度K（K）基于离散最小二乘问题的多项式${<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}}{<trans\; data-src="\sum ">\sum </trans>}_{<trans\; data-src="\ell ">Ş</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}{\left[\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans>\right]}^{\mathrm{<trans\; data-src="2">2</trans>}}$.英寸图1，我们给出了这样一个离散最小二乘拟合的示例。由此产生的近似误差$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="\Vert ">\Vert </trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}$对于$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="5">5</trans>}$为0.020，而5阶截断切比雪夫近似为0.347。尽管事实上${<trans\; data-src="sup">啜饮</trans>}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="[">[</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="]">]</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>$离散最小二乘近似等于0.650，而切比雪夫近似等于0.347。

虽然在我们的环境中，我们无法获得完整的特征值集，但我们在这项工作中的方法是利用最近的发展来有效估计矩阵的谱密度$\mathbf{<trans\; data-src="A">A类</trans>}$，使多项式适应谱，以便在（未知）特征值处获得更好的近似精度。在下一节回顾了谱密度估计之后，我们在第3节.英寸第4节，我们进行数值实验，近似$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$对于不同的矩阵$\mathbf{<trans\; data-src="A">A类</trans>}$和功能（f），并讨论建议的方法比最先进的方法工作得更好的情况。在第5节和第6节，我们分别探讨了该技术在局部图谱滤波器字典原子范数快速估计和时间顶点信号快速滤波中的应用。

2.光谱密度估算

这个累积谱密度函数或经验谱累积分布矩阵的$\mathbf{<trans\; data-src="A">A类</trans>}$定义为

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="N">N个</trans>}}\underset{<trans\; data-src="\ell ">Ş</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\mathrm{<trans\; data-src="1">1</trans>}{\mathrm{<trans\; data-src="1">1</trans>}}_{\left\{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="z">z</trans>}\right\}}<trans\; data-src=",">,</trans>\end{array}$$

(5)

哪里$\mathrm{<trans\; data-src="1">1</trans>}{\mathrm{<trans\; data-src="1">1</trans>}}_{<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="\}">\}</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$if语句C类是正确的，并且$\mathrm{<trans\; data-src="1">1</trans>}{\mathrm{<trans\; data-src="1">1</trans>}}_{<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="\}">\}</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$否则。这个谱密度函数[36]（第6章）（也称为状态密度或经验谱分布[37]（第2.4章）$\mathbf{<trans\; data-src="A">A类</trans>}$概率测度定义为

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="N">N个</trans>}}\underset{<trans\; data-src="\ell ">Ş</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\mathrm{<trans\; data-src="1">1</trans>}{\mathrm{<trans\; data-src="1">1</trans>}}_{\left\{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="z">z</trans>}\right\}}<trans\; data-src="·">·</trans>\end{array}$$

Lin等人[9]概述近似这些函数的方法。在这项工作中，我们使用了核多项式方法（KPM）的一种变体[38,39,40]中描述的[9,41]估计累积谱密度函数${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$属于$\mathbf{<trans\; data-src="A">A类</trans>}$即，对于每个S公司线性间隔点${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="S">S公司</trans>}}$之间$\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$和$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$，我们估计特征值的数量小于或等于${\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$通过随机轨迹估计[42,43]. 让$\mathbf{<trans\; data-src="x">x个</trans>}$表示具有分布的高斯随机向量$\mathcal{<trans\; data-src="N">N个</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathbf{<trans\; data-src="I">我</trans>}<trans\; data-src=")">)</trans>$,${<trans\; data-src="\{">\{</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="J">J型</trans>}}$表示样本大小J型来自该分布，以及${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}}_{{\mathsf{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$表示Jackson–Chebyshev多项式近似值${\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}_{{\mathsf{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}{\mathrm{<trans\; data-src="1">1</trans>}}_{\left\{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\right\}}$[44,45]. 特征值数目小于或等于的随机迹估计${\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$然后由给出

$${\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="tr">信托收据</trans>}\left({\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}_{{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\right)<trans\; data-src="=">=</trans>\mathbb{<trans\; data-src="E">E类</trans>}<trans\; data-src="[">[</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}_{{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="x">x个</trans>}<trans\; data-src="]">]</trans><trans\; data-src="\approx ">\approx </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="J">J型</trans>}}\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="J">J型</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}}_{{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=".">。</trans>$$

(6)

如中所示[46]，然后形成近似值${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$到${\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$通过执行单调分段三次插值[47]关于点的级数${\left\{\left({\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\frac{{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}{\mathrm{<trans\; data-src="N">N个</trans>}}\right)\right\}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="S">S公司</trans>}}$.分析差异化${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$产生近似值${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$到光谱密度函数${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$.自${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$是一个单调三次样条，我们也可以解析计算它的反函数${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$.我们在第3节利用两者${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$和${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans>$在生成多项式近似时，聚焦于特征值密度较高的谱区域。图2显示了八个实对称矩阵的估计累积谱密度函数的示例$\mathbf{<trans\; data-src="A">A类</trans>}$：Erdös的图Laplacians–Renyi图（gnp）图1明尼苏达州交通网络[48] ($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2642">2642</trans>}$)和斯坦福兔子图[49] ($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2503">2503</trans>}$); 随机传感器网络的归一化图Laplacian($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="5000">5000</trans>}$)来自[50]; 和网络25($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="9520">9520</trans>}$)，硅2($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="769">769</trans>}$)，笼架9($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3534">3534</trans>}$)和saylr4($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3564">3564</trans>}$)SuiteSparse矩阵集合中的矩阵[51]（我们使用$\frac{\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src="+">+</trans>{\mathbf{<trans\; data-src="A">A类</trans>}}^{<trans\; data-src="\top ">\top </trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}$对于cage9，以及net25和saylr4，我们基于以下非对角元素生成图Laplacians$\mathbf{<trans\; data-src="A">A类</trans>}$对于saylr4，我们将整个拉普拉斯量表的因子$\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2000">2000</trans>}}$).

形成估计的计算复杂性${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$是$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}\mathrm{<trans\; data-src="J">J型</trans>}{\mathrm{<trans\; data-src="K">K（K）</trans>}}_{\mathrm{<trans\; data-src="\Theta ">\Theta </trans>}}<trans\; data-src=")">)</trans>$，其中M（M）是中非零项的数目$\mathbf{<trans\; data-src="A">A类</trans>}$,J型是中的随机向量数(6)（在我们的实验中，$\mathrm{<trans\; data-src="J">J型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$足够了），以及${\mathrm{<trans\; data-src="K">K（K）</trans>}}_{\mathrm{<trans\; data-src="\Theta ">\Theta </trans>}}$是Jackson–Chebyshev多项式近似的次数${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}}_{{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$[41]. 然而，如果计算$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$单人间（f）和一张单人床$\mathbf{<trans\; data-src="b">b条</trans>}$，每个只需计算一次$\mathbf{<trans\; data-src="A">A类</trans>}$如果对多个函数重复此计算（f）或多个矢量$\mathbf{<trans\; data-src="b">b条</trans>}$，就像上面提到的应用程序中经常出现的情况一样。

3.光谱自适应方法

在本节中，我们将介绍两个新的学位类别K（K）多项式近似${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$到$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$，两者都利用了估计的累积光谱密度函数${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$。

3.1. 光谱自适应多项式插值

在第一种方法中，我们取${\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\frac{<trans\; data-src="cos">余弦</trans>\left(\frac{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="\pi ">\pi </trans>}}{\mathrm{<trans\; data-src="K">K（K）</trans>}}\right)<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}$，用于$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}$，哪些是$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}$度的极值K（K）切比雪夫多项式移位到区间$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$然后，我们通过设置估计的累积光谱密度函数的倒数来扭曲这些点${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans>$，在找到唯一度之前K（K）通过点的多项式插值${<trans\; data-src="\{">\{</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}$扭曲插值点背后的直觉是（i）在具有更多插值点的谱（域）区域中获得更好的近似，（ii）(三)仅取决于特征值的误差A类，所以我们希望在具有许多特征值的区域中，近似效果最好，（iii）如所示图3使用估计的累积谱密度函数的逆函数作为翘曲函数，可以获得更高的翘曲点密度$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>$落在光谱的高密度区域$\mathbf{<trans\; data-src="A">A类</trans>}$因此，理想情况下，翘曲应能在频谱的高密度区域中产生更多插值点，更好地逼近这些区域中的目标函数，进而减少误差(三)与由点生成的插值相比，插值在光谱中分布更均匀。

查找（唯一）学位K（K）多项式插值，我们的数值实现使用MATLAB的拟合函数，该函数将数据集中并缩放，然后通过QR分解求解生成的方程组。一旦获得插值多项式系数，${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$可以计算，例如通过(2)或者将插值多项式表示为切比雪夫多项式的线性组合，并在相关的三项递推中使用切比雪夫系数[23,32]. 整个过程在算法1中有详细说明。

算法1光谱自适应多项式插值。

输入厄米矩阵$\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$，矢量$\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$，功能（f），多项式次数K（K）

输出度K（K）近似${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\approx ">\approx </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$

1: 计算${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$，累计光谱密度的估计$\mathbf{<trans\; data-src="A">A类</trans>}$ 2: 对于k个在里面$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=":">以下为：</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}$ 做 三：   ${\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\leftarrow ">\leftarrow </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="cos">余弦</trans><trans\; data-src="(">(</trans>\frac{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="\pi ">\pi </trans>}}{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="]">]</trans>$ 4:   ${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\leftarrow ">\leftarrow </trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans>$ 5: 结束 6: 找到（唯一的）学位K（K）多项式插值${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$通过这些点${<trans\; data-src="\{">\{</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}$ 7: 计算${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$通过(2) 第8页： 返回${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$

3.2. 谱自适应多项式回归/正交多项式展开

第二种方法是求解加权最小二乘多项式回归问题

$$\begin{array}{c}\hfill \underset{\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}}{<trans\; data-src="min">最小值</trans>}\underset{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="M">M（M）</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}{\left[\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>\right]}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\end{array}$$

(7)

横坐标在哪里${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}$和重量${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}$被选择来捕获估计的光谱密度函数。我们研究了几种设置点的方法（例如，线性间隔点、区间上的切比雪夫点$<trans\; data-src="[">[</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="]">]</trans>$，每个子区间上的切比雪夫点$<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="]">]</trans>$，并通过估计的累积谱密度函数的逆函数扭曲点，如第3.1节)和权重（例如，分析计算的估计值${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$在谱密度函数中，基于特征值计数的谱密度函数的离散估计(6)，基于截断切比雪夫展开的原始KPM状态密度方法[9]（方程式3.11），或翘曲点的同等重量）。在没有深入讨论这些不同选项的详细信息的情况下，我们注意到选择横坐标$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>$在求解加权最小二乘问题时，如果使用由估计密度函数的逆函数扭曲的点，则可能会出现彼此非常接近的情况，从而导致数值不稳定性。在数值实验中，我们使用M（M）间隔上间隔均匀的点$<trans\; data-src="[">[</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="]">]</trans>$（即。，${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="-">-</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$)，并将权重设置为${\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="=">=</trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>$.要解决(7)，我们使用MATLAB的自定义变体拟合该函数再次通过QR分解求解加权最小二乘问题的法线方程：

$$\begin{array}{c}\hfill {\mathbf{<trans\; data-src="\Psi ">\Psi </trans>}}_{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\top ">\top </trans>}\mathbf{<trans\; data-src="W">W公司</trans>}{\mathbf{<trans\; data-src="\Psi ">\Psi </trans>}}_{\mathbf{<trans\; data-src="x">x个</trans>}}\mathbf{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="\Psi ">\Psi </trans>}}_{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\top ">\top </trans>}\mathbf{<trans\; data-src="W">W公司</trans>}\mathbf{<trans\; data-src="y">年</trans>}<trans\; data-src=",">,</trans>\end{array}$$

哪里${\mathbf{<trans\; data-src="\Psi ">\Psi </trans>}}_{\mathbf{<trans\; data-src="x">x个</trans>}}$是与点关联的Vandermonde矩阵$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>$,$\mathbf{<trans\; data-src="W">W公司</trans>}$是一个对角线矩阵，其对角线元素等于权重$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>$,$\mathbf{<trans\; data-src="y">年</trans>}$是一个列向量，其条目等于$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>$、和$\mathbf{<trans\; data-src="c">c（c）</trans>}$是未知多项式系数的向量。一旦系数$\mathbf{<trans\; data-src="c">c（c）</trans>}$最优多项式的，${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}^{<trans\; data-src="*">*</trans>}$获得，${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$可以再次计算，例如，通过(2). 算法2中详细介绍了这种方法的总结（正如一位匿名评论员所指出的，因为我们的估计${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$谱密度函数是一个分段二次函数，优化问题(7)可以被其连续模拟所取代，${<trans\; data-src="min">最小值</trans>}_{\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}}{<trans\; data-src="\int ">\int </trans>}_{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}}^{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}}{\left[\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\right]}^{\mathrm{<trans\; data-src="2">2</trans>}}{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}$，可以对许多函数进行解析求解$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans><trans\; data-src="·">·</trans><trans\; data-src=")">)</trans>$).

算法2频谱自适应多项式回归。

输入厄米矩阵$\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$，矢量$\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$，函数（f），多项式次数K（K），网格点数M（M）

输出度K（K）近似${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\approx ">\approx </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$

1: 计算${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$，光谱密度的估计$\mathbf{<trans\; data-src="A">A类</trans>}$ 2: 对于米在里面$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=":">以下为：</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}$ 做 三：   ${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\leftarrow ">\leftarrow </trans>\frac{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="-">-</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$ 4:   ${\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\leftarrow ">\leftarrow </trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>$ 5: 结束 6: ${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="\leftarrow ">\leftarrow </trans>{<trans\; data-src="argmin">argmin（最小值）</trans>}_{\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="\in ">\in </trans>{\mathcal{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}}{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}{\left[\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>\right]}^{\mathrm{<trans\; data-src="2">2</trans>}}$ 7: 计算${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$通过(2) 第8页： 返回${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$

查看此加权最小二乘法的另一种方法[52]是关于离散测度正交多项式的截断展开式$\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}$在点处有有限支撑$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>$和关联的内积[53]（第1.1节）

$$\begin{array}{c}\hfill {<trans\; data-src="\langle ">\langle </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}<trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathbb{<trans\; data-src="R">R（右）</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="M">M（M）</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">。</trans>\end{array}$$

这个M（M）离散一元正交多项式${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$满足三项递推关系[53]（第1.3节）

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\end{array}$$

(8)

具有${\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$,${\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}$,

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans>\frac{{<trans\; data-src="\langle ">\langle </trans>\mathrm{<trans\; data-src="x">x个</trans>}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}}{{<trans\; data-src="\langle ">\langle </trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="and">和</trans>}{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans>\frac{{<trans\; data-src="\langle ">\langle </trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}}{{<trans\; data-src="\langle ">\langle </trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">。</trans>\end{array}$$

给定横坐标$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>$和重量$<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="w">w个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans>$，三项递归系数${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$和${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$也可以通过稳定的Lanczos类型算法在$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$矩阵[53]（第2.2.3节）[54]. 在矩阵-向量表示法中，向量${\mathbf{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="M">M（M）</trans>}}$，它们是在M（M）横坐标，可以通过以下关系迭代计算

$$\begin{array}{c}\hfill {\mathbf{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="diag">诊断</trans>}<trans\; data-src="(">(</trans><trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\}">\}</trans><trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathbf{<trans\; data-src="I">我</trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathbf{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\end{array}$$

具有${\mathbf{<trans\; data-src="\pi ">\pi </trans>}}_{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="0">0</trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}$和${\mathbf{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="1">1</trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}$。图4显示了这些离散正交多项式的示例。

最后，学位K（K）多项式逼近$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$计算为

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="K">K（K）</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\frac{{<trans\; data-src="\langle ">\langle </trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}}{{<trans\; data-src="\langle ">\langle </trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="\rangle ">\rangle </trans>}_{\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="M">M（M）</trans>}}}}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>\end{array}$$

具有${\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="0">0</trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}}$,${\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans>\mathbf{<trans\; data-src="b">b条</trans>}$、和

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathbf{<trans\; data-src="I">我</trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="where">哪里</trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">。</trans>\end{array}$$

在进行数值实验之前，我们简要评论了本节中提出的谱自适应近似与Lanczos近似之间的关系$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$，由给出[19]（第13.2节）[23]

$$\begin{array}{c}\hfill {\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}^{<trans\; data-src="\top ">\top </trans>}\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>\end{array}$$

(9)

哪里${\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$是一个$\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$其列构成正交基的矩阵${\mathcal{<trans\; data-src="K">K（K）</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="span">跨度</trans>}\left\{\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>\mathbf{<trans\; data-src="Ab">抗体</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="A">A类</trans>}}^{\mathrm{<trans\; data-src="K">K（K）</trans>}}\mathbf{<trans\; data-src="b">b条</trans>}\right\}$，Krylov子空间。在(9),${\mathbf{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}^{<trans\; data-src="\top ">\top </trans>}\mathbf{<trans\; data-src="A">A类</trans>}{\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$是一个$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\times ">\times </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$三对角雅可比矩阵${\mathbf{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$等于$\frac{\mathbf{<trans\; data-src="b">b条</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}$.近似值(9)也可以写成${\mathrm{<trans\; data-src="q">q个</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$，其中${\mathrm{<trans\; data-src="q">q个</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$是学位K（K）插值函数的多项式（f）在$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}$的特征值${\mathbf{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$[19]（定理13.5）[55]. 因此，与经典多项式近似方法（如截断Cheybshev展开）不同，Lanczos方法间接适用于$\mathbf{<trans\; data-src="A">A类</trans>}$Lanczos方法与建议方法的不同之处在于${\mathbf{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$和Lanczos逼近多项式${\mathrm{<trans\; data-src="q">q个</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$取决于初始向量$\mathbf{<trans\; data-src="b">b条</trans>}$具体来说，多项式$<trans\; data-src="\{">\{</trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>$由表单的三项循环生成(8)

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\pi ">\pi </trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

使用${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}$和${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}$对角线和超对角线项的系数${\mathbf{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}$分别与分段常数测度正交

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left\{\begin{array}{cc}\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\hfill & \mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\hfill \\ \underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="i">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{<trans\; data-src="[">[</trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="b">b条</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\hfill & {\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}\hfill \\ \underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{<trans\; data-src="[">[</trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="b">b条</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\hfill & {\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="x">x个</trans>}\hfill \end{array}\right.<trans\; data-src=",">,</trans>\end{array}$$

哪里$\stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="b">b条</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="V">五</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathbf{<trans\; data-src="q">q个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="V">五</trans>}}^{<trans\; data-src="\top ">\top </trans>}\left(\frac{\mathbf{<trans\; data-src="b">b条</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}\right)$、和$\stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="b">b条</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>$是它的吗j个th分量[56]（定理4.2）。如果$\stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="b">b条</trans>}}$恰好是一个常量向量，那么$\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="P">P（P）</trans>}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$来自(5). 如果$\mathbf{<trans\; data-src="A">A类</trans>}$是图拉普拉斯算子，$\stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="b">b条</trans>}}$是图形傅里叶变换[4]第页，共页$\mathbf{<trans\; data-src="b">b条</trans>}$，归一化为单位能量。

4.数值示例和讨论

在图5，用于不同的功能$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$和矩阵$\mathbf{<trans\; data-src="A">A类</trans>}$，我们近似$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$具有$\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="=">=</trans>\mathbf{<trans\; data-src="V">五</trans>}\mathbf{<trans\; data-src="1">1</trans>}$和多项式近似阶从$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}$至25。估算累积光谱密度函数${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$带参数$\mathrm{<trans\; data-src="S">S公司</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$,$\mathrm{<trans\; data-src="J">J型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$、和${\mathrm{<trans\; data-src="K">K（K）</trans>}}_{\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="30">30</trans>}$，我们使用KPM，如所示图2基于的解析导数和反函数${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$，我们获得了两个建议的谱自适应多项式近似$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans>$，在计算每个${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$通过相应的三项递归。我们将所提出的方法与截断Chebyshev展开和具有相同多项式阶数的Lanczos方法进行了比较。请注意，当$\mathbf{<trans\; data-src="b">b条</trans>}$是光谱域中的常数向量$\mathbf{<trans\; data-src="A">A类</trans>}$，相对误差$\frac{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}{{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}$等于$\frac{{<trans\; data-src="\sum ">\sum </trans>}_{<trans\; data-src="\ell ">Ş</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}{\left(\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}}{{<trans\; data-src="\sum ">\sum </trans>}_{<trans\; data-src="\ell ">Ş</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="\ell ">Ş</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}$，其分子是中提到的离散最小二乘目标第1节。的第一列图5显示所有特征值的错误$\mathbf{<trans\; data-src="A">A类</trans>}$每阶10次多项式近似$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$第二列检查近似中相对误差的收敛性${\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathbf{<trans\; data-src="A">A类</trans>}}\mathbf{<trans\; data-src="b">b条</trans>}$对于具有不同谱分布的矩阵，对于四种方法中的每一种。

在图形信号处理领域[4]，分析或修改图形信号 $\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$，其中$\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>$是图形信号在顶点的值我加权连通图的G公司具有N个顶点，通过应用图形光谱过滤器${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}$.过滤的信号正是产品(1)图的拉普拉斯函数（或其他对称矩阵）和向量的函数，图信号$\mathbf{<trans\; data-src="b">b条</trans>}$.英寸图6，我们展示了此类函数集合的示例，通常称为图谱滤波器组。在右侧的三列中图5，我们检查了近似值产生的相对误差${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$用于低通、带通和高通图频谱滤波器${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$,${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}$、和${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="5">5</trans>}}$如所示图6a.我们扩展了这种图谱滤波器在第5节。

我们根据数值示例进行了两次观察：

• 谱自适应插值方法通常适用于低阶近似($\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="10">10</trans>}$)，但由于多项式插值对特定$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}$插值点（即插值点高度振荡）。
• 对于具有大量不同内部特征值的矩阵，如si2和cage9，所提出的谱自适应加权最小二乘法往往优于Lanczos方法。

5.应用一：局部化图谱滤波器字典原子的范数估计

从赋权无向图上的数据中提取信息的一种常见方法是将图信号表示为称为原子的积木图信号的线性组合，其集合称为字典。在本节中，我们考虑具有以下形式的局部频谱图滤波器字典$\mathcal{<trans\; data-src="D">D类</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="\{">\{</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=";">；</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="L">L（左）</trans>}}$，其中每个原子定义为${\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$具有$\mathbf{<trans\; data-src="L">L（左）</trans>}$拉普拉斯图和${\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$顶点值为1我其他地方为0。每个原子都可以解释为由滤波函数表征的光谱模式局部化的结果${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}$以顶点为中心我在图表中。请参见[58]有关局部谱图滤波器字典及其在无数信号处理和机器学习任务中作为变换和正则化器的应用的更多详细信息，请参阅。

对于大型稀疏图，字典原子从不显式计算；相反，它们与图形信号的内积近似为$<trans\; data-src="\langle ">\langle </trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="\rangle ">\rangle </trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}<trans\; data-src="\approx ">\approx </trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$，使用多项式近似方法，如第1节然而，在图形信号处理应用中，如阈值去噪或压缩[58,59,60]或图形信号的非均匀随机采样和插值[41,45,58,61]，快速估计字典原子的规范通常很重要，$<trans\; data-src="\{">\{</trans><trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}$.自$<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="\langle ">\langle </trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=",">,</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="\rangle ">\rangle </trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是该类型的双线性形式${\mathbf{<trans\; data-src="u">单位</trans>}}^{<trans\; data-src="\top ">\top </trans>}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="u">单位</trans>}$，单个原子的范数可以通过Lanczos求积等求积方法来估计[53]（第3.1.7章）[56]（第7章）[62]; 然而，为所有人这样做$\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="L">L（左）</trans>}$原子不可计算，因为每个原子都需要不同的函数和起始向量组合。其他替代方法包括[63]用于估计矩阵的对角元素，该矩阵不明确可用，但矩阵向量乘积易于计算，如$<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\sqrt{{\left[{\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>\right]}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}}}$。

在这个应用示例中，我们通过矩阵函数与随机向量的乘积来估计字典原子的范数，如下所示。让$\mathbf{<trans\; data-src="x">x个</trans>}$是一个随机向量，每个分量都具有独立且相同的标准正态分布（事实上，我们只是利用随机分量具有单位方差的性质）。那么我们有

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="var">无功功率，无功功率</trans>}\left(<trans\; data-src="\langle ">\langle </trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=",">,</trans>\mathbf{<trans\; data-src="x">x个</trans>}<trans\; data-src="\rangle ">\rangle </trans>\right)<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="var">无功功率，无功功率</trans>}\left(\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=")">)</trans>\right)<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="var">无功功率，无功功率</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=".">。</trans>\end{array}$$

因此，要估计$<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}$，估计就足够了$\mathrm{<trans\; data-src="sd">标准偏差</trans>}\left(<trans\; data-src="\langle ">\langle </trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=",">,</trans>\mathbf{<trans\; data-src="x">x个</trans>}<trans\; data-src="\rangle ">\rangle </trans>\right)<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="sd">标准偏差</trans>}\left({\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="x">x个</trans>}\right)<trans\; data-src="\approx ">\approx </trans>\mathrm{<trans\; data-src="sd">标准偏差</trans>}<trans\; data-src=" "> </trans>\left({\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="x">x个</trans>}\right)$获得学位K（K）多项式近似${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}$到${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}$因此，我们将每个原子范数估计定义为样本标准偏差

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="\nu ">\nu </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="sd">标准偏差</trans>}\left({\left\{{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right\}}_{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="J">J型</trans>}}\right)<trans\; data-src="\approx ">\approx </trans><trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\end{array}$$

（10）

其中每个${\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}$是随机向量的实现$\mathbf{<trans\; data-src="x">x个</trans>}$在数值实验中，我们比较了切比雪夫多项式近似和谱自适应加权最小二乘多项式近似的估计。除计算外，在通过KPM估算光谱密度的过程中(6)，我们已经计算了${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="T">T型</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}$对于每个$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}$和每个j个，其中${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="T">T型</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}$切比雪夫多项式是否移到区间$<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{<trans\; data-src="max">最大值</trans>}<trans\; data-src="]">]</trans>$根据这些量，我们可以很容易地计算${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="L">L（左）</trans>}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}$中的向量(10)对于不同的值我（不同的过滤器）。请参见[41]（第III.B.1节）。

在的顶行图7，我们演示了通过在图6b到兔子图的每个顶点。图7a显示了$\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="L">L（左）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2503">2503</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10012">10012</trans>}$字典原子。在图7b、 我们绘制原子的估计范数，通过度生成$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="8">8</trans>}$谱自适应加权最小二乘多项式逼近$\mathrm{<trans\; data-src="J">J型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="50">50</trans>}$中的随机向量(10)与实际原子规范相反。估计的原子范数与实际范数之比如所示图7c.我们显示了相对误差的平均值$\left|\frac{{\mathrm{<trans\; data-src="\nu ">\nu </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathit{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}\right|$作为一个单一的点跨越所有这些原子图7d、 并用不同的K（K）和J型和不同类的逼近多项式，以及图7e–f。在所有三个图上，在J型，用于低度数K（K）与基于切比雪夫多项式近似的估计相比，谱自适应多项式最小二乘法的估计具有更低的平均相对误差。虽然这些示例是在中小型图上进行的，以便与精确的原子规范进行比较，但该方法可以有效地扩展到为大型稀疏图设计的字典。

6.应用二：时间-频率信号的快速滤波

在本节中，我们演示了在第3节加速时域和图谱域中时间-顶点信号的联合滤波[50,64,65].

6.1. Time-Vertex信号

我们考虑一个加权的无向图$\mathrm{<trans\; data-src="G">G公司</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathcal{<trans\; data-src="V">五</trans>}<trans\; data-src=",">,</trans>\mathcal{<trans\; data-src="E">E类</trans>}<trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="W">W公司</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="\}">\}</trans>$具有N个顶点，其中$\mathcal{<trans\; data-src="V">五</trans>}$是顶点集，$\mathcal{<trans\; data-src="E">E类</trans>}$是一组边，并且${\mathbf{<trans\; data-src="W">W公司</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$是加权邻接矩阵。组合图Laplacian定义为${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="-">-</trans>{\mathbf{<trans\; data-src="W">W公司</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$，其中${\mathbf{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$与对角线${\mathbf{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>$等于我第个顶点。在每个顶点，我们观察到T型观察。因此，时变图形信号可以表示为矩阵$\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="T">T型</trans>}}$，其中${\mathbf{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}}$是上的值我第个顶点位于j个第次。图8显示了时间顶点信号的示例。

6.2. Time-Vertex过滤

确定时间点，每列$\mathbf{<trans\; data-src="X">X（X）</trans>}$是在上定义的图形信号G公司.让${\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="1">1</trans>}}$表示j个第列，共列$\mathbf{<trans\; data-src="X">X（X）</trans>}$,$\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}$.基于的图形结构G公司，我们可以在${\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}$例如滤波、去噪、修复和压缩[4]. 特别是对于过滤器$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src=":">以下为：</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="C">C类</trans>}$关于特征值的定义${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$，图谱滤波${\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}$可以计算为$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\Lambda ">\Lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}^{<trans\; data-src="*">*</trans>}{\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="t">t吨</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}$，其中${\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}^{<trans\; data-src="*">*</trans>}$是的共轭转置${\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$、和${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}{\mathrm{<trans\; data-src="\Lambda ">\Lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}^{<trans\; data-src="*">*</trans>}$是的光谱分解${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$。

相反，聚焦于G公司，的我第行，共行$\mathbf{<trans\; data-src="X">X（X）</trans>}$是离散时间信号${\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}^{<trans\; data-src="\top ">\top </trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="T">T型</trans>}}$，它指示信号值如何随时间变化我第个顶点。我们可以计算${\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}^{<trans\; data-src="\top ">\top </trans>}$通过${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="x">x个</trans>}}}_{{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}^{<trans\; data-src="\top ">\top </trans>}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="x">x个</trans>}}_{{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}^{<trans\; data-src="\top ">\top </trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathbf{<trans\; data-src="U">U型</trans>}}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$，其中${\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$是规格化DFT矩阵T型、和${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathbf{<trans\; data-src="U">U型</trans>}}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$是它的复共轭。DFT将信号从时域转换为频域，并允许信号的不同频率分量的放大或衰减。此过程在经典信号处理中称为频率滤波[4]. 经典一维信号的频率滤波等效于环形图上图形信号的图形谱滤波[66]（定理5.1）。让${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$用表示环图的图LaplacianT型顶点。其谱分解给出${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}{\mathrm{<trans\; data-src="\Lambda ">\Lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{<trans\; data-src="*">*</trans>}$，其中${\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$包括DFT基向量（归一化为长度为1）$\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\mathrm{<trans\; data-src="h">小时</trans>}$特征值由下式给出${\mathrm{<trans\; data-src="\Lambda ">\Lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="cos">余弦</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\pi ">\pi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="T">T型</trans>}}$，用于$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}$。

时变图形信号的联合时顶傅里叶变换$\mathbf{<trans\; data-src="X">X（X）</trans>}$定义为图形傅里叶变换和DFT的组合[50]:

$$\begin{array}{c}\hfill \stackrel{<trans\; data-src="^">^</trans>}{\mathbf{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}^{<trans\; data-src="*">*</trans>}\mathbf{<trans\; data-src="X">X（X）</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathbf{<trans\; data-src="U">U型</trans>}}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=".">。</trans>\end{array}$$

联合时间-顶点过滤器$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src=":">以下为：</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathbb{<trans\; data-src="C">C类</trans>}$为以下所有组合定义$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>$哪里${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$是的特征值${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$和${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$是的特征值${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$。联合时间-顶点过滤相应地定义为

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src=":">以下为：</trans><trans\; data-src="=">=</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="\circ ">\circ </trans><trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}^{<trans\; data-src="*">*</trans>}\mathbf{<trans\; data-src="X">X（X）</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathbf{<trans\; data-src="U">U型</trans>}}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{<trans\; data-src="\top ">\top </trans>}<trans\; data-src=",">,</trans>\end{array}$$

(11)

哪里$\mathbf{<trans\; data-src="H">H（H）</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="T">T型</trans>}}$有个条目${\mathbf{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{{\mathrm{<trans\; data-src="R">R（右）</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}<trans\; data-src=")">)</trans>$，∘表示两个矩阵的入口乘积。图9显示了两个时间-顶点滤波器示例：理想低通滤波器

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}{\mathrm{<trans\; data-src="1">1</trans>}}_{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="<"><</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}}_{<trans\; data-src="max">最大值</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\end{array}$$

(12)

和滤波器

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="5">5</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="100">100</trans>}{\left|<trans\; data-src="arccos">电弧炉</trans>\left(\frac{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}\right)<trans\; data-src="-">-</trans><trans\; data-src="arccos">电弧炉</trans>\left(\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\frac{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}}{{\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}}_{<trans\; data-src="max">最大值</trans>}}\right)\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src=",">,</trans>\end{array}$$

(13)

哪里${{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}}_{<trans\; data-src="max">最大值</trans>}$是的最大特征值${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$.的特征值${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$范围从0到4，无论大小T型，所以$\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}}_{<trans\; data-src="max">最大值</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}$。

作为谱滤波的二维扩展，时间顶点滤波将输入信号分解为$\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="T">T型</trans>}$正交分量，其中每个分量对应于图的拉普拉斯特征向量和DFT基函数的外积。然后，这些分量被相应的标量放大或衰减$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>$最后，将缩放后的分量相加，得到滤波后的信号。

6.3. 时间顶点滤波的频谱自适应逼近

由于计算所需的谱分解的高度复杂性${\mathbf{<trans\; data-src="U">U型</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$，已经开发了近似方法来加速联合时间-顶点滤波，例如Chebyshev2D[64]，ARMA2D[67]，以及快速傅立叶-切比雪夫算法[50].

如算法3所述，我们可以使用第3节有效地逼近时间顶点信号的滤波。我们方法的总体复杂性是$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="log">日志</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=")">)</trans>$，其中M（M）是中非零项目的数量${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$.的FFTN个离散时间长度信号T型取得$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="log">日志</trans>\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src=")">)</trans>$.循环计算T型具有次多项式的矩阵函数的谱自适应逼近K（K），因此具有以下复杂性$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="T">T型</trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=")">)</trans>$用于稀疏${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$。

算法3光谱自适应近似时间-顶点过滤。

输入无向赋权图G公司具有N个顶点，时间顶点信号$\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="T">T型</trans>}}$，过滤器小时

输出时间-顶点滤波信号$\mathbf{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="T">T型</trans>}}$

1: $\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="X">X（X）</trans>}}<trans\; data-src="\leftarrow ">\leftarrow </trans>$的FFT$\mathbf{<trans\; data-src="X">X（X）</trans>}$，其中n个第行，共行$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="X">X（X）</trans>}}$是的1D FFTn个第行，共行$\mathbf{<trans\; data-src="X">X（X）</trans>}$，用于$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}$ 2: 估计G公司 三： 对于t吨在里面$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=":">以下为：</trans>\mathrm{<trans\; data-src="T">T型</trans>}$ 做 4:   ${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{{\mathrm{<trans\; data-src="R">R（右）</trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}}<trans\; data-src=")">)</trans>$ 5:   查找最佳订单K（K）多项式近似${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans>$到${\mathrm{<trans\; data-src="f">（f）</trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans>$通过在扭曲的切比雪夫点上进行插值（如第3.1节)，或使用等距横坐标和估计光谱PDF中的权重进行加权最小二乘回归（如第3.2节) 6:   计算${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="x">x个</trans>}}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}$，其中${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="x">x个</trans>}}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}$是$\mathrm{<trans\; data-src="t">t吨</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\mathrm{<trans\; data-src="h">小时</trans>}$第列，共列$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="X">X（X）</trans>}}$ 7:   $\mathrm{<trans\; data-src="t">t吨</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\mathrm{<trans\; data-src="h">小时</trans>}$第列，共列$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="Y">Y（Y）</trans>}}$$<trans\; data-src="\leftarrow ">\leftarrow </trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=")">)</trans>{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="x">x个</trans>}}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}$ 第8页： 结束 9: $\mathbf{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src="\leftarrow ">\leftarrow </trans>$的逆FFT$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="Y">Y（Y）</trans>}}$，其中n个第行，共行$\mathbf{<trans\; data-src="Y">Y（Y）</trans>}$是的1D逆FFTn个第行，共行$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathbf{<trans\; data-src="Y">Y（Y）</trans>}}$ 10: 返回$\mathbf{<trans\; data-src="Y">Y（Y）</trans>}$

6.4. 数值实验

我们考虑理想的低通滤波器(12)和滤波器(13). 我们近似$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}$对于两个过滤器函数，使用$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1000">1000</trans>}$观察值，对于不同的图形G公司：gnp($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="500">500</trans>}$)，说4($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3564">3564</trans>}$)和随机传感器网络($\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="5000">5000</trans>}$)，其累积光谱密度如所示图2。在每种情况下，我们都会选择$\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="T">T型</trans>}}}{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=";">；</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="T">T型</trans>}}{\mathbf{<trans\; data-src="V">五</trans>}}_{{\mathrm{<trans\; data-src="G">G公司</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}{\mathbf{<trans\; data-src="V">五</trans>}}_{{\mathrm{<trans\; data-src="T">T型</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}}^{<trans\; data-src="\top ">\top </trans>}$即，联合谱域中的常数向量${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}$和${\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}$为了测试不同特征值对组合下近似方法的平均性能。多项式近似阶从$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$到30，我们按照算法3中描述的过程来近似$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}$.我们估计了累积谱密度函数${\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="P">P（P）</trans>}}}_{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src=")">)</trans>$带参数$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$,$\mathrm{<trans\; data-src="J">J型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}$、和${\mathrm{<trans\; data-src="K">K（K）</trans>}}_{\mathsf{<trans\; data-src="\Theta ">\Theta </trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="30">30</trans>}$。我们使用$\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="100">100</trans>}$在谱自适应多项式回归/正交多项式展开时寻找最佳多项式逼近。我们将所提出的方法与截断Chebyshev展开和具有相同多项式阶数的Lanczos方法进行了比较。对于每种方法，我们检验了Frobenius范数中相对误差的收敛性K（K）对于具有各种结构的图（以及拉普拉斯特征值的各种分布）。结果总结如下图10与我们在图5，我们看到谱自适应插值方法在较低多项式阶数下表现良好$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="5">5</trans>}<trans\; data-src=")">)</trans>$，但在较高的多项式阶数下往往不稳定。

6.5. 动态网格去噪

最后，我们复制了一个动态网格去噪示例，如[50]（第VI.B节）。走狗三维网格的原始时变序列特征网格来自$\mathrm{<trans\; data-src="T">T型</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="59">59</trans>}$时间实例，每个实例都具有$\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2502">2502</trans>}$三维空间中的点（x-y-z坐标）。此序列由$\mathrm{<trans\; data-src="2502">2502</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="59">59</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="3">三</trans>}$矩阵$\mathbf{<trans\; data-src="X">X（X）</trans>}$原始3D网格$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="5">5</trans>}$如所示图11a.添加高斯噪声（平均值0，标准偏差等于$\frac{{\mathrm{<trans\; data-src="0.2">0.2</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="F">F类</trans>}}}{\sqrt{\mathrm{<trans\; data-src="2502">2502</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="59">59</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="3">三</trans>}}}$)到每个网格点坐标。我们表示噪声三维网格序列，其中一个元素如所示图11b、 由$\mathbf{<trans\; data-src="Y">Y（Y）</trans>}$。通过减去平均值来创建单个图形x个,年、和z该网格中每个噪波网格的坐标，在所有59个时间实例中平均中心噪波网格坐标，然后在平均中心噪波网格坐标上构建5个最近邻图。结果图及其光谱密度函数如所示图11e–f。通过求解Tikhonov正则化问题对动态网格序列进行去噪[50]（方程式（30））

$$\begin{array}{c}\hfill {\mathbf{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="denoised">去噪的</trans>}}<trans\; data-src="=">=</trans>\underset{\mathbf{<trans\; data-src="Z">Z轴</trans>}}{<trans\; data-src="argmin">argmin（最小值）</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathbf{<trans\; data-src="Z">Z轴</trans>}<trans\; data-src="-">-</trans>{\mathbf{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="F">F类</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="tr">信托收据</trans>}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="Z">Z轴</trans>}}^{<trans\; data-src="\top ">\top </trans>}{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}\mathbf{<trans\; data-src="Z">Z轴</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="tr">信托收据</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="Z">Z轴</trans>}{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}{\mathbf{<trans\; data-src="Z">Z轴</trans>}}^{<trans\; data-src="\top ">\top </trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">。</trans>\end{array}$$

(14)

优化问题(14)有一个封闭的解决方案

$$\begin{array}{c}\hfill {\mathbf{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="denoised">去噪的</trans>}}<trans\; data-src="(">(</trans><trans\; data-src=":">以下为：</trans><trans\; data-src=",">,</trans><trans\; data-src=":">以下为：</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="tik">提克</trans>}}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathbf{<trans\; data-src="L">L（左）</trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="Y">Y（Y）</trans>}<trans\; data-src="(">(</trans><trans\; data-src=":">以下为：</trans><trans\; data-src=",">,</trans><trans\; data-src=":">以下为：</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=";">；</trans>\end{array}$$

(15)

即，中定义的联合时间顶点过滤操作(11)使用相同的联合不可分离低通滤波器，应用于每个有噪声的x、y和z坐标${\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="tik">提克</trans>}}$，在联合谱域中定义为[50]（方程式（31））

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="tik">提克</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}}<trans\; data-src=".">。</trans>\end{array}$$

(16)

我们执行网格搜索以查找参数值${\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$和${\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$使相对误差最小化$\frac{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>{\mathbf{<trans\; data-src="X">X（X）</trans>}}_{\mathrm{<trans\; data-src="denoised">去噪的</trans>}}<trans\; data-src="-">-</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="F">F类</trans>}}}{{<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>\mathbf{<trans\; data-src="X">X（X）</trans>}<trans\; data-src="|">|</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="F">F类</trans>}}}$对噪声的多次实现进行平均。在图11g、 我们显示了结果过滤器${\mathrm{<trans\; data-src="h">小时</trans>}}_{\mathrm{<trans\; data-src="tik">提克</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="G">G公司</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="R">R（右）</trans>}}<trans\; data-src=")">)</trans>$具有${\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="7.20">7.20</trans>}$和${\mathrm{<trans\; data-src="\tau ">\tau </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.45">0.45</trans>}$关于图的联合谱图11e.中的黑色虚线图11h表示中精确计算的去噪序列之间的相对误差(15)以及原始动态3D网格序列，平均超过20个加性高斯噪声实现。同一图像中的其他三条曲线显示了计算时的平均相对误差(15)通过快速傅里叶-切比雪夫方法进行近似[50]Chebyshev2D方法[64]，以及算法3的频谱自适应近似时间顶点滤波（快速傅里叶加权最小二乘），针对不同的多项式次数值K（K）所有三种近似都收敛到精确解和相应的误差，但程度较低($\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="10">10</trans>}$)，光谱自适应快速傅里叶加权最小二乘法产生更好的近似值。图11c–d一次显示去噪网格示例($\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="5">5</trans>}$)由其中两个近似值得出$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="6">6</trans>}$两者之间的差异很细微，但可以通过扫描狗的顶部看到，网格点形成了稍微光滑的表面。

7.结论

我们提出了两种新的频谱自适应多项式逼近方法$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>\mathbf{<trans\; data-src="b">b条</trans>}$对于大型稀疏矩阵：谱自适应插值和谱自适应加权最小二乘/正交多项式展开。这些方法利用矩阵的累积谱密度的估计来构建固定阶的多项式K（K）得出更好的近似值（f）在矩阵谱的高密度区域。在逼近精度方面，数值实验表明，相对于最先进的Lanczos和Chebysev多项式逼近技术，所提出的方法通常在较低的多项式阶数下获得更精确的逼近；然而，所提出的频谱自适应插值方法在较高程度上不是很稳定($\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="10">10</trans>}$)由于过盈。所提出的谱自适应加权最小二乘法在具有许多不同内部特征值的矩阵的精度方面表现得特别好，而Lanczos方法，例如，在K（K）更高和/或矩阵$\mathbf{<trans\; data-src="A">A类</trans>}$具有许多极值特征值。一个潜在的扩展是研究一种结合Lanczos和光谱自适应加权最小二乘法的混合方法。关于函数形状的相对近似精度，我们没有注意到一致的趋势（f）。

就计算复杂性而言，我们的方法（如切比雪夫多项式近似）的成本为$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>$，其中$\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="z">z</trans>}<trans\; data-src="(">(</trans>\mathbf{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>$。对于非常大的稀疏矩阵，此复杂性降低为$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>$，其中$\mathbf{<trans\; data-src="A">A类</trans>}$是一个$\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}$另一方面，Lanczos方法需要额外的$\mathcal{<trans\; data-src="O">O（运行）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}{\mathrm{<trans\; data-src="K">K（K）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$正交化步骤导致的成本，使得足够大的成本更高K（K）最后，所提出的频谱自适应方法，如切比雪夫近似，能够通过相邻节点之间的通信进行高效的并行和分布式计算[33]. 另一方面，Lanczos方法的内部产品可能会在某些分布式计算环境（例如GPU）中导致额外的通信费用或严重的效率损失。