Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning

TL;DR

This paper introduces a low-rank factorization approach for graph learning from data, enabling efficient estimation of graph structures in high-dimensional settings by leveraging Riemannian optimization techniques.

Contribution

It proposes a novel low-rank constrained framework for graph learning that scales better with data dimension and applies Riemannian optimization to solve the resulting problems.

Findings

Efficient dimension-versus-performance trade-off demonstrated on synthetic data.

Low-rank approach outperforms traditional methods in large-scale scenarios.

Method effective on real data, confirming practical applicability.

Abstract

This paper addresses the problem of learning an undirected graph from data gathered at each nodes. Within the graph signal processing framework, the topology of such graph can be linked to the support of the conditional correlation matrix of the data. The corresponding graph learning problem then scales to the squares of the number of variables (nodes), which is usually problematic at large dimension. To tackle this issue, we propose a graph learning framework that leverages a low-rank factorization of the conditional correlation matrix. In order to solve for the resulting optimization problems, we derive tools required to apply Riemannian optimization techniques for this particular structure. The proposal is then particularized to a low-rank constrained counterpart of the GLasso algorithm, i.e., the penalized maximum likelihood estimation of a Gaussian graphical model. Experiments on synthetic and real data evidence that a very efficient dimension-versus-performance trade-off can be achieved with this approach.