The Hodge Laplacian in higher-order data analysis

Graph theory is a well-established and widely used framework for modeling problems in data analysis, and one of the most helpful tools in this context is the graph Laplacian. The capacity to represent pairwise relations is indeed crucial, but is often not enough. If the underlying structure of the problem deals with relations between more than two entities, or even if the information we are interested in flows on the edges rather than on the vertices, we need a more complex instrument to properly capture these nuances. For this reason we introduce simplicial complexes, a higher-order generalization of the concept of graph that can encode more of the subtleties of the problem at hand. As simplicial complexes act as the natural evolution of graphs, the Hodge Laplacian replaces the usual graph Laplacian as the corresponding operator acting on simplicial complexes. This operator captures relations of simplicial complexes of different dimensions, extending the concept of adjacency that is classically only defined between vertices and edges. To illustrate the potential of the Hodge Laplacian, we show data analysis applications in which the limits faced by graphs and graph Laplacian are overcome through the use of this new, more sophisticated operator.