The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression

Abstract:Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.

Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2607.24041 [math.ST]

(or arXiv:2607.24041v1 [math.ST] for this version)

https://doi.org/10.48550/arXiv.2607.24041 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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