Physics-Informed Singular-Value Learning for Cross-Covariances Forecasting in Financial Markets

2026-01-12

AI Summary

This work addresses the limitations of existing random matrix theory (RMT)-based cross-covariance cleaning methods, which often fail to deliver robust out-of-sample predictions in real-world financial markets characterized by non-stationarity and pervasive global factors. To overcome this, we propose a novel neural network architecture inspired by RMT that learns an end-to-end nonlinear mapping from observed to cleaned singular values within the empirical singular vector basis. Our approach uniquely integrates physical priors from RMT directly into the network design, preserving the structural optimality of theoretical solutions while retaining the flexibility to adapt to dynamic market conditions. Empirical evaluations on long-horizon equity return data demonstrate that the proposed model significantly outperforms purely analytical methods, achieving lower prediction error and a superior bias-variance trade-off.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Web data integration and cleaning
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

A new wave of work on covariance cleaning and nonlinear shrinkage has delivered asymptotically optimal analytical solutions for large covariance matrices. The same framework has been generalized to empirical cross-covariance matrices, whose singular value decomposition identifies canonical comovement modes between two asset sets, with singular values quantifying the strength of each mode and providing natural targets for shrinkage. Existing analytical cross-covariance cleaners are derived under strong stationarity and large-sample assumptions, and they typically rely on mesoscopic regularity conditions such as bounded spectra; macroscopic common modes (e.g., a global market factor) violate these conditions. When applied to real equity returns, where dependence structures drift over time and global modes are prominent, we find that these theoretically optimal formulas do not translate into robust out-of-sample performance. We address this gap by designing a random-matrix-inspired neural architecture that operates in the empirical singular-vector basis and learns a nonlinear mapping from empirical singular values to their corresponding cleaned values. By construction, the network can recover the analytical solution as a special case, yet it remains flexible enough to adapt to non-stationary dynamics and mode-driven distortions. Trained on a long history of equity returns, the proposed method achieves a more favorable bias-variance trade-off than purely analytical cleaners and delivers systematically lower out-of-sample cross-covariance prediction errors. Our results demonstrate that combining random-matrix theory with machine learning makes asymptotic theories practically effective in realistic time-varying markets.

Problem

• cross-covariance forecasting
• non-stationarity
• global market factor
• singular-value shrinkage
• covariance cleaning

Innovation

• physics-informed learning
• singular-value shrinkage
• cross-covariance forecasting
• random matrix theory
• non-stationary financial markets