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.

翻译：我们研究了处理定义在抽象拓扑空间（建模为单纯复形，可解释为节点、边、三角面等构成的图的推广）上信号的线性滤波器。为处理此类信号，我们开发了单纯复形卷积滤波器，定义为下霍奇拉普拉斯矩阵和上霍奇拉普拉斯矩阵的矩阵多项式。首先，我们研究了这些滤波器的性质，证明它们是线性和平移不变的，同时具有置换和方向等变性。这些滤波器还可以以低计算复杂度分布式实现，因为它们仅涉及（多轮）上下相邻单纯复形之间的单纯移位。其次，以边流为研究对象，我们分析了这些滤波器的频率响应，并探讨如何利用霍奇分解来划分梯度、旋度和调和频率。我们讨论了这些频率如何分别对应下相邻耦合、上相邻耦合以及霍奇拉普拉斯矩阵的核，并且可以通过我们的滤波器设计独立调节。第三，我们研究了设计单纯复形卷积滤波器的不同流程，并讨论了它们的相对优势。最后，我们在多个应用中验证了我们的单纯复形滤波器：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。