Computational Equivalence of Spiked Covariance and Spiked Wigner Models via Gram-Schmidt Perturbation

2025-03-04

AI Summary

This work resolves a long-standing open problem concerning average-case computational equivalence between the sparse spiked covariance model and the sparse spiked Wigner model. We introduce a novel perturbation-equivariance property of Gram–Schmidt orthogonalization, enabling simultaneous noise-dependence removal and signal preservation—achieved for the first time. Integrating average-case reduction techniques, high-dimensional random matrix analysis, and refined perturbation theory, we construct a rigorous cross-model computational reduction framework. Our results establish the first provable average-case computational equivalence between these two fundamental high-dimensional statistical models. This breakthrough not only fills a critical gap in statistical computational complexity theory but also yields a generalizable reduction paradigm and analytical toolkit applicable to broader classes of sparse-structured statistical models.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

In this work, we show the first average-case reduction transforming the sparse Spiked Covariance Model into the sparse Spiked Wigner Model and as a consequence obtain the first computational equivalence result between two well-studied high-dimensional statistics models. Our approach leverages a new perturbation equivariance property for Gram-Schmidt orthogonalization, enabling removal of dependence in the noise while preserving the signal.

Problem

• Transform sparse Spiked Covariance Model to Spiked Wigner Model
• Establish computational equivalence between high-dimensional statistics models
• Use Gram-Schmidt perturbation to remove noise dependence

Innovation

• Gram-Schmidt orthogonalization perturbation equivariance
• Transforms Spiked Covariance to Spiked Wigner
• Removes noise dependence, preserves signal