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

TL;DR

The paper addresses cross-covariance estimation in high-dimensional, time-varying financial data where macroscopic market modes violate the assumptions of analytical shrinkage. It introduces a physics-informed neural architecture that preserves rotational invariance and operates in the empirical singular-vector basis, learning a nonlinear map from the spectrum and marginal projections to cleaned values, with BBP shrinkage recovered as a limiting case. The approach is dimension-agnostic and uses a two-stream tokenization plus a bidirectional LSTM to produce residual corrections, yielding cleaned singular values $ ilde{s}_k$ via $ ilde{s}_k = ar{s}_k + ilde{}$ (or a bounded multiplicative form) for $k le r$. Empirical results on synthetic benchmarks demonstrate compatibility with BBP in stationary regimes and robustness to market-mode perturbations, while real-market validation shows systematic out-of-sample gains over BBP and MLE, supported by diagnostics that confirm feasibility and non-stationarity-driven improvements. Overall, the work shows that integrating random-matrix theory with machine learning yields practically effective cross-covariance denoising and forecasting in time-varying markets.

Abstract

A new wave of work on covariance cleaning and nonlinear shrinkage has delivered asymptotically optimal analytical solutions for large covariance matrices. Building on this progress, these ideas have been generalized to empirical cross-covariance matrices, whose singular-value shrinkage characterizes comovements between one set of assets and another. 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.

Figures (4)

• Figure 1: Flowchart of the neural singular value cleaning architecture. The diagram illustrates the construction of dual-stream tokens from marginal projections $\overline{\gamma}$ and singular values $\overline{s}$, their transformation through a shared encoder $E_{\theta}$, and the global context aggregation via a bidirectional LSTM and pointwise head $g_\theta$ to produce the final additive corrections $\delta_k$.
• Figure 2: OOS MSE for cross-correlation estimators on financial data covering 2017–2024. For each test window, the estimator is trained on an expanding sample from 1995 up to the year immediately preceding the test period, and the reported value averages 1,000 independent runs for each out-of-sample year. The figures report also the Oracle (red dashed line), which is the lower achievable MSE by a RIE (see eq. \ref{['eq:oracle']}). The left panels preserve the chronological order between in-sample and out-of-sample windows, whereas the right panels shuffle dates. The upper panels varies the total number of assests for $\nu=0.25$, the lower panels varies the relative dimension $\nu$ for $n=1000$. The error bars indicate a $95\%$ percentile $100,000$ copies bootstrap confidence interval.
• Figure 3: OOS MSE for cross-correlation estimators on financial data covering 2017–2024 with the market mode removed. For each test window, the estimator is trained on an expanding sample from 1995 up to the year immediately preceding the test period, and the reported value averages 1,000 independent runs for each OOS year. The left panels preserve the chronological order between in-sample and OOS windows, whereas the right panels shuffle dates. The upper panels vary the total number of assets for $\nu=0.25$, while the lower panels vary the relative dimension $\nu$ for $n=1000$. The error bars indicate a $95\%$ bootstrap confidence interval.
• Figure 4: Feasibility diagnostic for the reconstructed cross--correlation block. Top row: empirical distributions of canonical singular values (singular values of the whitened block $\widetilde{\mathbf{C}}^{(w)}_{XY}$ obtained via Eq. \ref{['eq:whitened_def']}) for the original chronological pipeline (left) and the shuffled control (right). Bottom row: cleaned singular values in the unwhitened domain plotted against the empirical singular values, again for the original pipeline (left) and the shuffled control (right).