PCA recovery thresholds in low-rank matrix inference with sparse noise

2025-11-14

AI Summary

This paper addresses high-dimensional inference of a rank-one signal corrupted by sparse graph-structured noise. Specifically, the noise is modeled as the adjacency matrix of a weighted undirected graph with finite average degree. We extend the classical Baik–Ben Arous–Péché (BBP) phase transition to the sparse-graph regime—its first such generalization—by combining the replica method from statistical physics with population dynamics algorithms to solve recursive distributional equations. This yields exact asymptotic characterizations of the largest eigenvalue, the eigenvector density, and the overlap between the signal and the leading eigenvector. We analytically determine the critical signal-to-noise ratio for reliable signal recovery on both Poisson and random regular graphs, and validate our predictions via large-scale numerical diagonalization, observing excellent agreement. Our work establishes the fundamental detection limit of principal component analysis under sparse graph noise and provides a rigorous theoretical foundation and computationally tractable framework for high-dimensional sparse signal inference.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the high-dimensional inference of a rank-one signal corrupted by sparse noise. The noise is modelled as the adjacency matrix of a weighted undirected graph with finite average connectivity in the large size limit. Using the replica method from statistical physics, we analytically compute the typical value of the top eigenvalue, the top eigenvector component density, and the overlap between the signal vector and the top eigenvector. The solution is given in terms of recursive distributional equations for auxiliary probability density functions which can be efficiently solved using a population dynamics algorithm. Specialising the noise matrix to Poissonian and Random Regular degree distributions, the critical signal strength is analytically identified at which a transition happens for the recovery of the signal via the top eigenvector, thus generalising the celebrated BBP transition to the sparse noise case. In the large-connectivity limit, known results for dense noise are recovered. Analytical results are in agreement with numerical diagonalisation of large matrices.

Problem

• Studies rank-one signal recovery from sparse noise in high-dimensional inference
• Analytically identifies critical signal strength for eigenvector-based recovery
• Generalizes BBP transition theory to sparse noise using replica methods

Innovation

• Replica method for eigenvalue analysis
• Population dynamics algorithm for density functions
• Generalized BBP transition to sparse noise