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This paper reframes classical covariance shrinkage as empirical risk minimization over stochastic interpolants, identifying three specific mechanisms—scheduling, flow map couplings, and early stopping—to reduce statistical risk. By utilizing non-linear flow maps, the approach enables eigenvector regularization independent of the empirical estimate's eigenbasis, offering a novel neural estimator validated on neuroimaging data. The work provides a theoretical foundation for improving high-dimensional covariance estimation through optimal transport-inspired structures and approximation error bounds.
Covariance Shrinkage via Stochastic Interpolation
arXiv:2606.07382v1 Announce Type: new Abstract: We recast classical shrinkage of high-dimensional covariance estimators as empirical risk minimization over a parametric stochastic interpolant between a source and a target distribution. This formalism recovers known shrinkage estimators as special cases and reveals three distinct mechanisms for reducing statistical risk: (i) Scheduling: the interpolant schedule determines the class of admissible covariances, and hence the achievable risk. (ii) Flow maps and couplings: whereas naive constructions amount to assuming independence between the distributions, specific coupling structures (e.g., solutions of optimal transport problems) can lower the empirical risk. Moreover, non-linear flow maps realizing such couplings free the interpolant covariance from the eigenbasis of the empirical estimate, enabling eigenvector regularization. (iii) Early stopping: estimators defined by integrating a regressed vector field afford an additional bias-variance trade-off through approximation of the true interpolant distribution. We then propose a neural estimator of the interpolant, together with an upper bound on its quadratic risk in terms of the interpolant approximation error, and validate both on synthetic experiments. Finally, we apply the estimator to real neuroimaging data, demonstrating the additional regularization power this approach offers in practice.
Abstract:We recast classical shrinkage of high-dimensional covariance estimators as empirical risk minimization over a parametric stochastic interpolant between a source and a target distribution. This formalism recovers known shrinkage estimators as special cases and reveals three distinct mechanisms for reducing statistical risk: (i) Scheduling: the interpolant schedule determines the class of admissible covariances, and hence the achievable risk. (ii) Flow maps and couplings: whereas naive constructions amount to assuming independence between the distributions, specific coupling structures (e.g., solutions of optimal transport problems) can lower the empirical risk. Moreover, non-linear flow maps realizing such couplings free the interpolant covariance from the eigenbasis of the empirical estimate, enabling eigenvector regularization. (iii) Early stopping: estimators defined by integrating a regressed vector field afford an additional bias-variance trade-off through approximation of the true interpolant distribution. We then propose a neural estimator of the interpolant, together with an upper bound on its quadratic risk in terms of the interpolant approximation error, and validate both on synthetic experiments. Finally, we apply the estimator to real neuroimaging data, demonstrating the additional regularization power this approach offers in practice.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2606.07382 [cs.LG]
(or arXiv:2606.07382v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.07382 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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