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

arXiv: https://arxiv.org/abs/2607.24041 · HTML: https://mlsanity.com/paper/2607.24041/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 55 references that are in the map.
- First posted on arXiv 27 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [Generalization error of min-norm interpolators in transfer learning](https://mlsanity.com/paper/2406.13944/) (2024)
- [High-dimensional analysis of ridge regression for non-identically distributed data with a variance profile](https://mlsanity.com/paper/2403.20200/) (2024)
- [Resolvent convergence under heterogeneous second-moment profiles and quadratic-form control](https://mlsanity.com/paper/2109.02644/) (2021)
- [Gaussian and Non-Gaussian Universality of Data Augmentation](https://mlsanity.com/paper/2202.09134/) (2022)
- [High-dimensional ridge regression with random features for non-identically distributed data with a variance profile](https://mlsanity.com/paper/2504.03035/) (2025)

## Abstract (arXiv)

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.