# The Impact of Anisotropic Covariance Structure on the Training Dynamics and Generalization Error of Linear Networks

arXiv: https://arxiv.org/abs/2601.06961 · HTML: https://mlsanity.com/paper/2601.06961/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 40 references that are in the map.
- First posted on arXiv 11 Jan 2026.

## Builds on (most cited first, 30 of 30)

- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [Neural Networks as Kernel Learners: The Silent Alignment Effect](https://mlsanity.com/paper/2111.00034/) (2021)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [On the Optimal Weighted ℓ2 Regularization in Overparameterized Linear Regression](https://mlsanity.com/paper/2006.05800/) (2020)
- [More Than a Toy: Random Matrix Models Predict How Real-World Neural Representations Generalize](https://mlsanity.com/paper/2203.06176/) (2022)
- [An analytic theory of generalization dynamics and transfer learning in deep linear networks](https://mlsanity.com/paper/1809.10374/) (2018)
- [A Unifying View on Implicit Bias in Training Linear Neural Networks](https://mlsanity.com/paper/2010.02501/) (2020)
- [Asymptotics of Ridge (less) Regression under General Source Condition](https://mlsanity.com/paper/2006.06386/) (2020)
- [Width Provably Matters in Optimization for Deep Linear Neural Networks](https://mlsanity.com/paper/1901.08572/) (2019)
- [Scaling and renormalization in high-dimensional regression](https://mlsanity.com/paper/2405.00592/) (2024)
- [From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks](https://mlsanity.com/paper/2409.14623/) (2024)
- [Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning](https://mlsanity.com/paper/2406.06158/) (2024)
- [A Theory of Neural Tangent Kernel Alignment and Its Influence on Training](https://mlsanity.com/paper/2105.14301/) (2021)
- [The Asymmetric Maximum Margin Bias of Quasi-Homogeneous Neural Networks](https://mlsanity.com/paper/2210.03820/) (2022)
- [Deep linear networks for regression are implicitly regularized towards flat minima](https://mlsanity.com/paper/2405.13456/) (2024)
- [Theory IIIb: Generalization in Deep Networks](https://mlsanity.com/paper/1806.11379/) (2018)

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

- [To Use or not to Use Muon: How Simplicity Bias in Optimizers Matters](https://mlsanity.com/paper/2603.00742/) (2026)
- [Optimal Learning Rate Scaling Depends on Data in Deep Scalar Linear Networks](https://mlsanity.com/paper/2607.07884/) (2026)

## Abstract (arXiv)

The success of deep neural networks largely depends on the statistical structure of the training data. While learning dynamics and generalization on isotropic data are well-established, the impact of pronounced anisotropy on these crucial aspects is not yet fully understood. We examine the impact of data anisotropy, represented by a spiked covariance structure, a canonical yet tractable model, on the learning dynamics and generalization error of a two-layer linear network in a linear regression setting. Our analysis reveals that the learning dynamics proceed in two distinct phases, governed initially by the input-output correlation and subsequently by other principal directions of the data structure. Furthermore, we derive an analytical expression for the generalization error, quantifying how the alignment of the spike structure of the data with the learning task improves performance. Our findings offer deep theoretical insights into how data anisotropy shapes the learning trajectory and final performance, providing a foundation for understanding complex interactions in more advanced network architectures.