Robust Spectral Detection of Global Structures in the Data by Learning a Regularization

TL;DR

This paper introduces a learning-based regularization approach to improve spectral methods for detecting global structures in sparse or noisy data matrices, overcoming localization issues and achieving near-theoretical detection limits.

Contribution

The authors propose a novel method to learn regularization matrices that mitigate eigenvector localization, enhancing spectral detection in challenging data conditions.

Findings

Effective in community detection, clustering, rank estimation, and matrix completion.

Outperforms existing spectral algorithms in sparse, noisy scenarios.

Works close to theoretical detectability limits in synthetic data.

Abstract

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we propose a general method to solve the localization problem by learning a regularization matrix from the localized eigenvectors. Using matrix perturbation analysis, we demonstrate that the learned regularizations suppress down the eigenvalues associated with localized eigenvectors and enable us to recover the informative eigenvectors representing the global structure. We show applications of our method in several inference problems: community detection in networks, clustering from pairwise similarities, rank estimation and matrix completion problems. Using extensive experiments, we illustrate that our method solves the localization problem and works down to the theoretical detectability limits in different kinds of synthetic data. This is in contrast with existing spectral algorithms based on data matrix, non-backtracking matrix, Laplacians and those with rank-one regularizations, which perform poorly in the sparse case with noise.