Simplicial Convolutional Filters

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.

Topological Slepians: Maximally Localized Representations of Signals Over Simplicial Complexes

“…The size of the dictionary built in (13) can be controlled either tuning K d and K u , or by taking just the top K Slepians per each pair of concentration sets. Clearly, the maximum possible number of Slepians depends on the chosen sets {S, F}, because they affect the rank of the operator BF CS BF in (12). Therefore, for a given K and pair of concentration sets {S, F}, the number of topological Slepians will be min{ K, rank{BF CS BF }}.…”

“…Then, several papers have given important contributions to TSP. For instance, the work in [12] proposed FIR filters for signals defined over simplicial complexes, hinging on the Hodge decomposition, where the Fourier modes are eigenvectors of higher order combinatorial Laplacians [13]. The work in [14] introduced generalized Laplacian for embedding simplicial complexes into traditional graphs.…”

Simplicial Vector Autoregressive Model For Streaming Edge Flows

“…We can process a simplicial signal with simplicial convolutional filters, which for edge signals has the form [20,23]…”

“…The regularizer consists of three terms: i) ftL1,ℓ ft, where ft = Ftβ, ii) ftL1,u ft, and iii) P p=1 ∥β p ∥2. Here, i) and ii) impose constraints based on the simplicial structure, e.g., ftL1l ft = ∥B1 ft∥ flows at the nodes, and ftL1u ft = ∥B ⊤ 2 ft∥ 2 regularizes the cyclic flows (B ⊤ 2 ft) [12,20]. The third one is group-lasso regularizer, acting along domain by imposing group sparsity on {β p } P p=1 .…”

“…A simplicial complex is a mathematical representation of the higher-order connectivities of the networks, and allows for mathematical tractability in a way akin to graphs for pairwise similarities [18]. Although, the popular signal processing concepts such as the Fourier transform, sampling theory [19], and convolution filters [20] are developed recently for the simplexes, they can be seen as solutions to process time-invariant data, and ignore the spatio-temporal coupling present in simplicial processes.…”