# Risk Phase Transitions in Spiked Regression: Alignment Driven Benign and Catastrophic Overfitting

arXiv: https://arxiv.org/abs/2510.01414 · HTML: https://mlsanity.com/paper/2510.01414/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2025.

## Built on by (most cited first, 2 of 2)

- [Detecting overfitting in Neural Networks during long-horizon grokking using Random Matrix Theory](https://mlsanity.com/paper/2605.12394/) (2026)
- [High-dimensional ridgeless least squares interpolation under spiked covariance structures](https://mlsanity.com/paper/2608.07281/) (2026)

## Abstract (arXiv)

This paper analyzes the generalization error of minimum-norm interpolating solutions in linear regression using spiked covariance data models. The paper characterizes how varying spike strengths and target-spike alignments can affect risk, especially in overparameterized settings. The study presents an exact expression for the generalization error, leading to a comprehensive classification of benign, tempered, and catastrophic overfitting regimes based on spike strength, the aspect ratio c=d/n (particularly as c → ∞), and target alignment. Notably, in well-specified aligned problems, increasing spike strength can surprisingly induce catastrophic overfitting before achieving benign overfitting. The paper also reveals that target-spike alignment is not always advantageous, identifying specific, sometimes counterintuitive, conditions for its benefit or detriment. Alignment with the spike being detrimental is empirically demonstrated to persist in nonlinear models.