Topological Signal Processing Over Cell MultiComplexes Via Cross-Laplacian Operators

Abstract: One of the key challenges in many research fields is uncovering how different interconnected systems interact within complex networks, typically represented as multi-layer networks. Capturing the intra- and cross-layer interactions among different domains for analysis and processing calls for topological algebraic descriptors capable of localizing the homologies of different domains, at different scales, according to the learning task. Our first contribution in this paper is to introduce the Cell MultiComplexes (CMCs), which are novel topological spaces that enable the representation of higher-order interactions among interconnected cell complexes. We introduce cross-Laplacian operators as powerful algebraic descriptors of CMC spaces able to capture different topological invariants, whether global or local, at different resolutions. Using the eigenvectors of these operators as bases for the signal representation, we develop topological signal processing tools for signals defined over CMCs. Then, we focus on the signal spectral representation and on the filtering of noisy flows observed over the cross-edges between different layers of CMCs. We show that a local signal representation based on cross-Laplacians yields a better sparsity/accuracy trade-off compared to monocomplex representations, which provide overcomplete representation of local signals. Finally, we illustrate a topology learning strategy designed to infer second-order cross-cells between layers, with applications to brain networks for encoding inter-module connectivity patterns.