# On the Benefits of Weight Normalization for Overparameterized Matrix Sensing

arXiv: https://arxiv.org/abs/2510.01175 · HTML: https://mlsanity.com/paper/2510.01175/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 54 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 30 of 31)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)
- [Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent](https://mlsanity.com/paper/2005.08898/) (2020)
- [Global Convergence of Gradient Descent for Asymmetric Low-Rank Matrix Factorization](https://mlsanity.com/paper/2106.14289/) (2021)
- [Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing](https://mlsanity.com/paper/2301.11500/) (2023)
- [The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensing](https://mlsanity.com/paper/2302.01186/) (2023)
- [On the computational and statistical complexity of over-parameterized matrix sensing](https://mlsanity.com/paper/2102.02756/) (2021)
- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [Over-Parameterization Exponentially Slows Down Gradient Descent for Learning a Single Neuron](https://mlsanity.com/paper/2302.10034/) (2023)
- [The effect of smooth parametrizations on nonconvex optimization landscapes](https://mlsanity.com/paper/2207.03512/) (2022)
- [Implicit Regularization and Convergence for Weight Normalization](https://mlsanity.com/paper/1911.07956/) (2019)
- [Understanding Deflation Process in Over-parametrized Tensor Decomposition](https://mlsanity.com/paper/2106.06573/) (2021)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [How Over-Parameterization Slows Down Gradient Descent in Matrix Sensing: The Curses of Symmetry and Initialization](https://mlsanity.com/paper/2310.01769/) (2023)
- [On the Crucial Role of Initialization for Matrix Factorization](https://mlsanity.com/paper/2410.18965/) (2024)
- [Robust Implicit Regularization via Weight Normalization](https://mlsanity.com/paper/2305.05448/) (2023)
- [Convergence of Alternating Gradient Descent for Matrix Factorization](https://mlsanity.com/paper/2305.06927/) (2023)
- [Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization](https://mlsanity.com/paper/2504.09708/) (2025)
- [Factorization Approach for Low-complexity Matrix Completion Problems: Exponential Number of Spurious Solutions and Failure of Gradient Methods](https://mlsanity.com/paper/2110.10279/) (2021)
- [Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization](https://mlsanity.com/paper/2310.06159/) (2023)
- [Convergence of Gradient Descent with Small Initialization for Unregularized Matrix Completion](https://mlsanity.com/paper/2402.06756/) (2024)
- [Optimizing Orthogonalized Tensor Deflation via Random Tensor Theory](https://mlsanity.com/paper/2302.05798/) (2023)
- [Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixtures](https://mlsanity.com/paper/2506.06584/) (2025)

## Built on by (most cited first, 3 of 3)

- [Escaping Local Minima Provably in Non-convex Matrix Sensing: A Deterministic Framework via Simulated Lifting](https://mlsanity.com/paper/2602.05887/) (2026)
- [Low-Rank Adaptation Redux for Large Models](https://mlsanity.com/paper/2604.21905/) (2026)
- [Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics](https://mlsanity.com/paper/2606.31390/) (2026)

## Abstract (arXiv)

While normalization techniques are widely used in deep learning, their theoretical understanding remains relatively limited. In this work, we establish the benefits of (generalized) weight normalization (WN) applied to the overparameterized matrix sensing problem. We prove that WN with Riemannian optimization achieves linear convergence, yielding an exponential speedup over standard methods that do not use WN. Our analysis further demonstrates that both iteration and sample complexity improve polynomially as the level of overparameterization increases. To the best of our knowledge, this work provides the first characterization of how WN leverages overparameterization for faster convergence in matrix sensing.