Simplicial Convolutional Neural Networks

Abstract

Graphs can model networked data by representing them as nodes and their pairwise relationships as edges. Recently, signal processing and neural networks have been extended to process and learn from data on graphs, with achievements in tasks like graph signal reconstruction, graph or node classifications, and link prediction. However, these methods are only suitable for data defined on the nodes of a graph. In this paper, we propose a simplicial convolutional neural network (SCNN) architecture to learn from data defined on simplices, e.g., nodes, edges, triangles, etc. We study the SCNN permutation and orientation equivariance, complexity, and spectral analysis. Finally, we test the SCNN performance for imputing citations on a coauthorship complex.

Simplicial Convolutional Filters

“…Proof: See Appendix C. Intuitively, the two previous propositions state that for the simplicial filter H k the labeling and reference orientation of simplices are inconsequential for the filter output. These two properties have been previously reported in the context of neural network on SCs [27], [29], [32]. These equivariances imply that we can learn a filter to process a given simplex by seeing only permuted and reoriented versions of it: if two parts of an SC are topologically equivalent and the simplices support corresponding flows, a simplicial convolutional filter yields equivalent outputs.…”

“…Indeed, all the filters studied in the current paper can be generalized to cell complexes, by substituting the appropriate incidence matrices. Different neural network architectures on SCs have also been developed to learn from the simplicial data, e.g., [27], [28], [29], [30], [31], [32], [33]. Importantly, the linear operation in these different neural convolutional layers may be understood as a simplicial filter (as discussed here) with different filter parameters.…”

Autoregressive graph Volterra models and applications

“…The convex optimization problem (18) can be efficiently solved (and distributed) by off-the-shelf convex programming toolboxes. ( 16) X (1) = H (1) X (1) + H (2) X (2) + E (17)…”

Higher-Order Attention Networks

“…Recently, machine learning models have been developed to model data with support on topological spaces modeled as simplicial complexes [108,86,31,77,22,45,44,11], cell complexes [43,82,20,78] and hypergraphs [101,9,50,4,35,10,37]. These models have been used in link prediction [26,44], optimal homology generator detection [52], and trajectory prediction [77].…”