Simplicial Convolutional Filters

TL;DR

The paper addresses processing signals defined on simplicial complexes, extending graph signal processing to higher-order domains by introducing simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians: $\mathbf{H}_k = h_0 \mathbf{I} + \sum_{l_1=1}^{L_1} \alpha_{l_1} ( \mathbf{L}_{k,\ell})^{l_1} + \sum_{l_2=1}^{L_2} \beta_{l_2} ( \mathbf{L}_{k,\mathrm{u}})^{l_2}$. The framework leverages a shift-and-sum interpretation, enabling distributed implementation, and uses the Simplicial Fourier Transform (SFT) to analyze frequency components in the gradient, curl, and harmonic subspaces of edge signals. The paper makes three key contributions: (1) establishing linear, shift-invariant, permutation- and orientation-equivariant simplicial filters with decentralized shifting; (2) a spectral analysis showing independent control over gradient and curl frequencies via $\mathbf{L}_{k,\ell}$ and $\mathbf{L}_{k,\mathrm{u}}$ within the Hodge decomposition; and (3) three filter-design strategies—least-squares, grid-based universal design, and Chebyshev polynomial design—plus demonstrations in subcomponent extraction, edge-flow denoising, currency markets, and transportation networks. These results enable fast, scalable processing of higher-order signals onSCs and have practical impact for analyses of edge flows and higher-order relations in complex networks.

Abstract

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.

Figures (13)

• Figure 1: Simplicial Complexes and Signals. (a): An SC of order 2 containing seven nodes, ten edges and three 2-simplex (the shaded filled triangles). Reference orientations of the simplices are indicate by corresponding arrows (the reference orientation of a node is trivial). (b): An arbitrary edge flow, where a negative flow indicates that the actual flow direction is opposite to the reference orientation and the magnitude is denoted by the edge width.
• Figure 2: Simplicial shifting. (a): An edge flow indicator $\mathbf{f}$ of edge $\{5,6\}$. (b): One-step lower shifting $\mathbf{L}_{1,\ell} \mathbf{f}$. Edge $\{5,6\}$ and its direct lower neighboring edges (green) update their flows by aggregating information from their lower neighbors and themselves. (c): Two-step lower shifting $\mathbf{L}_{1,\ell}^2\mathbf{f}$. Lower neighboring edges (green) update their flows through faces within two hops away from edge $\{5,6\}$, which can be obtained by one-step shifting $\mathbf{L}_{1,\ell}\mathbf{f}_{\ell}$. (d): One-step upper shifting $\mathbf{L}_{1,\rm{u}}\mathbf{f}$. Edge $\{5,6\}$ and its upper neighbors (red) update their flows through local information aggregation. (e): Two-step upper shifting $\mathbf{L}_{1,\rm{u}}^2\mathbf{f}$. The output is localized within the one-hop upper neighborhood, as there is no upper neighboring edge two hops away from $\{5,6\}$. (f): Two-step shifting result $\mathbf{L}_1^2\mathbf{f}$, as the sum of (c) and (e).
• Figure 3: Simplicial convolutional filtering is a shift-and-sum operation.
• Figure 4: Flow decomposition illustration (all numbers rounded to two decimal places). (a): A synthetic edge flow $\mathbf{f}$. (b): The gradient component $\mathbf{f}_{\rm{G}}$ has a nonzero netflow at each node, but a zero flow around each triangle. (c): The curl component $\mathbf{f}_{\rm{C}}$ has a zero netflow at each node, i.e., is divergence-free, but a nonzero flow around each triangle. (d): The harmonic component $\mathbf{f}_{\rm{H}}$ has a zero netflow at each node and zero circulation around each triangle, i.e., is divergence- and curl-free (see Section \ref{['sec:subcomponent-extraction']}).
• Figure 5: Spectral analysis of the edge space in Fig. \ref{['1a']}. The flow value (all numbers rounded to two decimal places) is indicated by the edge width and annotated next to the edge. It is zero if the edge is not annotated. If a flow orientation is opposite to the reference orientation, the corresponding flow value is negative. (a)-(c): The 1st, 3rd, and 6th eigenvectors in the gradient space $\mathbf{U}_{\rm{G}}$ with the corresponding gradient frequencies $\lambda_{\rm{G}}$. The total divergence of the eigenvector increases with the eigenvalue. (d)-(e): The 1st and 3rd eigenvectors in the curl space $\mathbf{U}_{\rm{C}}$ with the corresponding curl frequencies $\lambda_{\rm{C}}$. The total curl of the eigenvectors increases with the eigenvalue. (f): The only eigenvector in the harmonic space $\mathbf{U}_{\rm{H}}$ has frequency 0 and zero divergence and curl.

Theorems & Definitions (9)

• Proposition 1
• Proposition 2: Permutation equivariance
• Proposition 3: Orientation equivariance
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Lemma 1
• Corollary 1