Real-world data, such as biological measurements, are naturally modeled using a graph, whose edges encode the pairwise relations between the measured signals. Traditionally, the graph underlying the data has been studied through the graph Laplacian, \(L\), and its spectrum. The eigenvalues and eigenvectors of the graph Laplacian encode valuable information about the intrinsic structure of the graph. For instance, the number of connected components of the graph is equal to the multiplicity of the zero-eigenvalue of \(L\). Recently, the spectrum of the graph Laplacian has been exploited for processing signals defined on graphs. A notion of frequency representation for graph signals was introduced, which made possible the definition of frequency filters, convolution, and other basic techniques in signal analysis.
In this talk I will explain how the theory of signal processing on graphs can be naturally extended to signal processing on simplicial complexes. The key ingredients are: the higher order combinatorial Laplacian, the well known combinatorial Hodge theorem and the new notion of higher order Laplacian Fourier transform, obtained by decomposing p-cochains into eigencochains of the Laplacian.
This is a joint work with Gard Spreemann, Michaël Defferrard and Kathryn Hess Bellwald.