Better together: Cross and joint covariances enhance signal detectability in undersampled data
Phys. Rev. E 114, 035305 – Published 17 September, 2026
DOI: https://doi.org/10.1103/fs5k-y33m
Abstract
Many data-science applications involve detecting a shared signal between two high-dimensional variables. Using random matrix theory methods, we determine when such a signal can be detected and reconstructed from sample correlations, despite the background of sampling-noise-induced correlations. We consider three different covariance matrices constructed from two high-dimensional variables: their individual self-covariance, their unwhitened cross-covariance as used in partial least squares (PLS), and the self-covariance of the concatenated (joint) variable, which incorporates the self- and the cross-correlation blocks. We observe the expected Baik, Ben Arous, and Péché detectability phase transition in all these covariance matrices, and we show that joint- and cross-covariance matrices always reconstruct the shared signal earlier than the self-covariances. Whether the joint or the cross approach is better depends on the mismatch of dimensionalities between the variables. We discuss what these observations mean for choosing the right method for detecting linear correlations in data and how these findings may generalize to nonlinear statistical dependencies.
Physics Subject Headings (PhySH)
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This article appears in the following collection:
Statistical Physics Meets Machine Learning - Machine Learning Meets Statistical Physics
The Editors of Physical Review E are pleased to present the Collection on Statistical Physics Meets Machine Learning - Machine Learning Meets Statistical Physics, highlighting research at the intersection of machine learning and statistical physics, on the occasion of the two Statistical Physics Meets Machine Learning and the two Machine Learning Meets Statistical Physics sessions at the 2025 Global Physics Summit. The Collection is being guest edited by David Schwab (CUNY, New York) and Yuhai Tu (IBM Watson Research Center Yorktown Heights, NY). Every article published in this collection underwent a rigorous peer review process, adhering to the same high standards applied to all papers. The Physical Review E editorial team managed the peer review and made all editorial decisions.
