单纯复形卷积滤波器
Simplicial Convolutional Filters
AI总结：
本文提出基于上下霍奇拉普拉斯算子矩阵多项式的单纯复形卷积滤波器，证明其线性、平移不变及置换与定向等变性，并利用霍奇分解独立调节梯度、旋度和谐波频率，在信号分解、边流去噪、金融市场和交通网络分析中验证了有效性。
摘要：
我们研究用于处理以单纯复形为模型的抽象拓扑空间上信号的线性滤波器，单纯复形可被解释为图的推广，能够表示节点、边、三角面等。为处理此类信号，我们开发了定义为下霍奇拉普拉斯算子和上霍奇拉普拉斯算子的矩阵多项式的单纯复形卷积滤波器。首先，我们研究这些滤波器的性质，并证明它们是线性的、平移不变的，并且具有置换等变性和定向等变性。这些滤波器还可以以分布式方式实现，且计算复杂度低，因为它们仅涉及上下相邻单纯形之间的（多轮）单纯形移位。其次，针对边流，我们研究这些滤波器的频率响应，并考察如何利用霍奇分解来划分梯度频率、旋度频率和谐波频率。我们讨论了这些频率分别如何对应于下相邻耦合、上相邻耦合以及霍奇拉普拉斯算子的核，并且可以通过我们的滤波器设计独立调节。第三，我们研究设计单纯复形卷积滤波器的不同方法，并讨论它们的相对优势。最后，我们在若干应用中验证了所提出的单纯复形滤波器：提取单纯复形信号的不同频率分量、对边流去噪，以及分析金融市场和交通网络。
英文摘要：
We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians. First, we study the properties of these filters and show that they are linear and shift-invariant, as well as permutation and orientation equivariant. These filters can also be implemented in a distributed fashion with a low computational complexity, as they involve only (multiple rounds of) simplicial shifting between upper and lower adjacent simplices. Second, focusing on edge-flows, we study the frequency responses of these filters and examine how we can use the Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We discuss how these frequencies correspond to the lower- and the upper-adjacent couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned independently by our filter designs. Third, we study different procedures for designing simplicial convolutional filters and discuss their relative advantages. Finally, we corroborate our simplicial filters in several applications: to extract different frequency components of a simplicial signal, to denoise edge flows, and to analyze financial markets and traffic networks.
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