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Physics-Informed Singular-Value Estimation for Cross-Covariance Cleaning
Efstratios Manolakis, University of Catania

ABSTRACT

This talk will introduce a cross-covariance estimator that is based on analytical Random Matrix Theory (RMT) [1] and extends it by using a physics-inspired neural network (PINN) [2] framework. Extending our previous work, we develop a physics-inspired architecture that blends denoising and forecasting to address complex, non-stationary dynamics beyond the scope of analytical theory. In the study of complex systems, characterizing the cross-covariance between distinct sets of variables is a fundamental challenge which suffers in high-dimensional regimes due to sampling noise. While recent progress in RMT has delivered asymptotically optimal analytical solutions [3, 4] for covariance cleaning, extending these results to the rectangular matrices remains an open question. Current research relies on strong stationarity and mesoscopic regularity conditions that are frequently violated in real-world data. Our method operates in the empirical singular-vector basis and defines a nonlinear mapping from empirical singular values to their cleaned counterparts. The architecture recovers the analytical RMT solution as a limiting case while utilizing a PINN to adapt to non-stationary distortions. We demonstrate that this framework systematically achieves lower out-of-sample mean squared errors than purely analytical cleaners across both synthetic benchmarks and decades of real-market equity returns.

REFERENCES

[1] Florent Benaych-Georges, Jean-Philippe Bouchaud, and Marc Potters. Optimal cleaning for singular values of cross-covariance matrices. The Annals of Applied Probability, 33(2):1295–1326, 2023.

[2] Christian Bongiorno, Efstratios Manolakis, and Rosario Nunzio Mantegna. End-to-end large portfolio optimization for variance minimization with neural networks through covariance cleaning. arXiv preprint arXiv:2507.01918, 2025.

[3] Olivier Ledoit and Michael Wolf. Quadratic shrinkage for large covariance matrices. Bernoulli, 28(3):1519–1547, 2022.

[4] Joël Bun, Jean-Philippe Bouchaud, and Marc Potters. Cleaning large correlation matrices: tools from random matrix theory. Physics Reports, 666:1–109, 2017.

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