Learning graphs and simplicial complexes from data

TL;DR

This paper introduces a novel method for learning both graph structures and three-node interactions (simplicial complexes) from data, using a structured Volterra framework and convex optimization, outperforming existing approaches.

Contribution

The paper presents a new approach that jointly infers graph topology and higher-order simplicial complexes from data, extending beyond pairwise interactions.

Findings

Superior performance on synthetic data

Effective identification of three-node interactions

Outperforms existing graph inference methods

Abstract

Graphs are widely used to represent complex information and signal domains with irregular support. Typically, the underlying graph topology is unknown and must be estimated from the available data. Common approaches assume pairwise node interactions and infer the graph topology based on this premise. In contrast, our novel method not only unveils the graph topology but also identifies three-node interactions, referred to in the literature as second-order simplicial complexes (SCs). We model signals using a graph autoregressive Volterra framework, enhancing it with structured graph Volterra kernels to learn SCs. We propose a mathematical formulation for graph and SC inference, solving it through convex optimization involving group norms and mask matrices. Experimental results on synthetic and real-world data showcase a superior performance for our approach compared to existing methods.