Learning the Topology of a Simplicial Complex Using Simplicial Signals: A Greedy Approach

TL;DR

This paper proposes a greedy, signal-based method to learn the topology of simplicial complexes, including higher-order interactions, from observed signals on nodes and edges, using a sparse and smoothness assumption.

Contribution

It introduces a novel nonconvex optimization approach and an efficient algorithm to identify the topology of simplicial complexes from signals, extending graph learning to higher-order structures.

Findings

Successfully learns higher-order simplices from signals

Efficient block-coordinate algorithm for topology inference

Demonstrates effectiveness on simulated data

Abstract

Graphs are ubiquitous to model the irregular (non-Euclidean) structure of complex data, but they are limited to pairwise relationships and fail to model the complexities of the datasets exhibiting higher-order interactions. In that context, simplicial complexes (SCs) are emerging as a tractable candidate to handle such domains. The first step in using SC-based processing and learning schemes is to identify the topology of the SC, which is the problem investigated in this paper. In particular, we assume that we observe a number of signals (features) associated with the nodes of the SC (simplices of order 0) as well as signals (features) associated with a subset of the edges of the SC (simplices of order 1). The goal is then to use these signals to learn the remaining edges as well as the triangles that are filled (simplices of order 2). To address this problem, we assume that the signals are smooth on the unknown SC and that the higher-order relations are sparse. We then postulate the learning problem as a nonconvex optimization and develop an efficient (block-coordinate) algorithm to identify the SC topology.