# Signal-Plus-Noise Decomposition of Nonlinear Spiked Random Matrix Models

arXiv: https://arxiv.org/abs/2405.18274 · HTML: https://mlsanity.com/paper/2405.18274/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 55 references that are in the map.
- First posted on arXiv 28 May 2024.

## Builds on (most cited first, 22 of 22)

- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Learning curves of generic features maps for realistic datasets with a teacher-student model](https://mlsanity.com/paper/2102.08127/) (2021)
- [The Gaussian equivalence of generative models for learning with shallow neural networks](https://mlsanity.com/paper/2006.14709/) (2020)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [The Benefits of Reusing Batches for Gradient Descent in Two-Layer Networks: Breaking the Curse of Information and Leap Exponents](https://mlsanity.com/paper/2402.03220/) (2024)
- [A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networks](https://mlsanity.com/paper/2310.07891/) (2023)
- [Asymptotics of feature learning in two-layer networks after one gradient-step](https://mlsanity.com/paper/2402.04980/) (2024)
- [Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models](https://mlsanity.com/paper/1807.00891/) (2018)
- [Spectral Evolution and Invariance in Linear-width Neural Networks](https://mlsanity.com/paper/2211.06506/) (2022)
- [What causes the test error? Going beyond bias-variance via ANOVA](https://mlsanity.com/paper/2010.05170/) (2020)
- [Nonlinear spiked covariance matrices and signal propagation in deep neural networks](https://mlsanity.com/paper/2402.10127/) (2024)
- [MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel](https://mlsanity.com/paper/1507.03857/) (2015)
- [A Random Matrix Perspective on Mixtures of Nonlinearities for Deep Learning](https://mlsanity.com/paper/1912.00827/) (2019)
- [Fundamental limits of detection in the spiked Wigner model](https://mlsanity.com/paper/1806.09588/) (2018)
- [Demystifying Disagreement-on-the-Line in High Dimensions](https://mlsanity.com/paper/2301.13371/) (2023)
- [Rank-one matrix estimation: analysis of algorithmic and information theoretic limits by the spatial coupling method](https://mlsanity.com/paper/1812.02537/) (2018)
- [Information-theoretic bounds for exact recovery in weighted stochastic block models using the Renyi divergence](https://mlsanity.com/paper/1509.06418/) (2015)
- [Optimal Algorithms for the Inhomogeneous Spiked Wigner Model](https://mlsanity.com/paper/2302.06665/) (2023)
- [Detection problems in the spiked matrix models](https://mlsanity.com/paper/2301.05331/) (2023)

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

- [On the Mechanisms of Weak-to-Strong Generalization: A Theoretical Perspective](https://mlsanity.com/paper/2505.18346/) (2025)
- [Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent](https://mlsanity.com/paper/2605.17767/) (2026)

## Abstract (arXiv)

In this paper, we study a nonlinear spiked random matrix model where a nonlinear function is applied element-wise to a noise matrix perturbed by a rank-one signal. We establish a signal-plus-noise decomposition for this model and identify precise phase transitions in the structure of the signal components at critical thresholds of signal strength. To demonstrate the applicability of this decomposition, we then utilize it to study new phenomena in the problems of signed signal recovery in nonlinear models and community detection in transformed stochastic block models. Finally, we validate our results through a series of numerical simulations.