Learning the Topology of a Simplicial Complex Using Noisy Simplicial Signals

Abstract

Graphs are a fundamental tool for modeling the irregular (non-Euclidean) structure of complex data. However, they are inherently limited to representing pairwise relationships, making them inadequate for datasets exhibiting higher-order interactions. Simplicial complexes (SCs) have emerged as a promising framework for capturing such higher-order dependencies. This paper focuses on the problem of identifying the topology of an SC from signals, which serves as the foundation for SC-based processing and learning schemes. We consider a setting where we observe noisy signals (features) associated with the nodes of the SC (0-simplices) and a subset of the edges (1-simplices). We assume the observed signals are smooth over the unknown SC topology, and that the higher-order interactions are sparse. Building on these assumptions, we formulate topology learning as a nonconvex optimization problem and propose an efficient block-coordinate descent (BCD) algorithm to solve it. A key step in our formulation is the modeling of the topology of the SC using binary edge and triangle selection vectors, combined with efficient greedy algorithms for optimizing such vectors. We establish theoretical convergence guarantees to a stationary point of a relaxed (penalized) version of the problem and discuss computational complexity. Multiple numerical experiments with both synthetic and real-world datasets validate the effectiveness of our approach, highlighting the capability of SC-learning methods to uncover and model higher-order relationships in complex datasets.