Simplicial Convolutional Filters

We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted generalizations of graphs that account nodes, edges, triangular faces, etc. To process such signals, we develop convolutional defined matrix polynomials the lower and upper Hodge Laplacians. First, properties these show they are shift-invariant, well permutation orientation equivariant. These can also implemented in a distributed fashion with low computational complexity, involve only (multiple rounds of) shifting between adjacent simplices. Second, focusing edge-flows, frequency responses examine how use Hodge-decomposition to delineate gradient, curl harmonic frequencies. discuss frequencies correspond lower- upper-adjacent couplings kernel Laplacian, respectively, tuned independently by our filter designs. Third, different procedures designing their relative advantages. Finally, corroborate several applications: extract components signal, denoise edge flows, analyze financial markets traffic networks.