Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Models

2025-06-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Zhenyu Liao and Michael W. Mahoney introduce an extended Random Matrix Theory (RMT) framework to analyze nonlinear Machine Learning (ML) and Deep Neural Networks (DNNs) in high-dimensional, overparameterized regimes. This work, submitted on June 16, 2025, and revised on April 16, 2026, moves beyond traditional eigenvalue-based RMT for linear models. The authors propose the "High-dimensional Equivalent" concept, which unifies Deterministic Equivalent and Linear Equivalent, to tackle challenges like high dimensionality, nonlinearity, and the analysis of generic eigenspectral functionals. This framework precisely characterizes training and generalization performance across linear models, shallow networks, and deep networks, explaining phenomena such as scaling laws, double descent, and nonlinear learning dynamics.

Key takeaway

For research scientists developing or deploying deep learning models in high-dimensional settings, this extended RMT framework offers a unified theoretical lens. You can use its insights to better understand and predict complex behaviors like double descent and scaling laws, informing model design and hyperparameter tuning. This theoretical foundation helps you navigate the counterintuitive aspects of overparameterized neural networks.

Key insights

Extended Random Matrix Theory unifies analysis of high-dimensional, nonlinear deep learning phenomena.

Principles

• Classical low-dimensional intuitions fail in overparameterized ML.
• High-dimensional Equivalent unifies RMT analysis for nonlinearity.

Method

The High-dimensional Equivalent framework systematically addresses high dimensionality, nonlinearity, and generic eigenspectral functional analysis to characterize DNN performance.

In practice

• Characterizes scaling laws in deep learning.
• Explains double descent phenomena.
• Analyzes nonlinear learning dynamics.

Topics

• Random Matrix Theory
• Deep Learning
• High-dimensional Data
• Overparameterized Models
• Generalization Performance

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.