Learning with latent group sparsity via heat flow dynamics on networks

TL;DR

This paper introduces a novel method for learning group structures in data using heat flow dynamics on networks, eliminating the need for prior group information and demonstrating theoretical guarantees and practical success across various domains.

Contribution

The authors develop a heat flow-based penalty method that leverages network geometry for learning group structures without prior labels, with proven performance bounds.

Findings

Effective learning with minimal heat flow time logarithmic in problem size.

No need for pre-processing clustering or spectral methods.

Successful real-world applications across diverse fields.

Abstract

Group or cluster structure on explanatory variables in machine learning problems is a very general phenomenon, which has attracted broad interest from practitioners and theoreticians alike. In this work we contribute an approach to learning under such group structure, that does not require prior information on the group identities. Our paradigm is motivated by the Laplacian geometry of an underlying network with a related community structure, and proceeds by directly incorporating this into a penalty that is effectively computed via a heat flow-based local network dynamics. In fact, we demonstrate a procedure to construct such a network based on the available data. Notably, we dispense with computationally intensive pre-processing involving clustering of variables, spectral or otherwise. Our technique is underpinned by rigorous theorems that guarantee its effective performance and provide bounds on its sample complexity. In particular, in a wide range of settings, it provably suffices to run the heat flow dynamics for time that is only logarithmic in the problem dimensions. We explore in detail the interfaces of our approach with key statistical physics models in network science, such as the Gaussian Free Field and the Stochastic Block Model. We validate our approach by successful applications to real-world data from a wide array of application domains, including computer science, genetics, climatology and economics. Our work raises the possibility of applying similar diffusion-based techniques to classical learning tasks, exploiting the interplay between geometric, dynamical and stochastic structures underlying the data.