# The spiked matrix model with generative priors

arXiv: https://arxiv.org/abs/1905.12385 · HTML: https://mlsanity.com/paper/1905.12385/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2019.

## Built on by (most cited first, 20 of 20)

- [Graph-based Approximate Message Passing Iterations](https://mlsanity.com/paper/2109.11905/) (2021)
- [Deterministic equivalent and error universality of deep random features learning](https://mlsanity.com/paper/2302.00401/) (2023)
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- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [Deep learning via message passing algorithms based on belief propagation](https://mlsanity.com/paper/2110.14583/) (2021)
- [Generative Principal Component Analysis](https://mlsanity.com/paper/2203.09693/) (2022)
- [Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity](https://mlsanity.com/paper/2106.11950/) (2021)
- [Optimal Algorithms for the Inhomogeneous Spiked Wigner Model](https://mlsanity.com/paper/2302.06665/) (2023)
- [Optimal Spectral Transitions in High-Dimensional Multi-Index Models](https://mlsanity.com/paper/2502.02545/) (2025)
- [Neural-prior stochastic block model](https://mlsanity.com/paper/2303.09995/) (2023)
- [Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models](https://mlsanity.com/paper/2403.03695/) (2024)
- [Asymptotic generalization error of a single-layer graph convolutional network](https://mlsanity.com/paper/2402.03818/) (2024)
- [Generalized Eigenvalue Problems with Generative Priors](https://mlsanity.com/paper/2411.01326/) (2024)
- [Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap](https://mlsanity.com/paper/2205.13527/) (2022)
- [Detection problems in the spiked matrix models](https://mlsanity.com/paper/2301.05331/) (2023)
- [Spatially heterogeneous learning by a deep student machine](https://mlsanity.com/paper/2302.07419/) (2023)

## Abstract (arXiv)

Using a low-dimensional parametrization of signals is a generic and powerful way to enhance performance in signal processing and statistical inference. A very popular and widely explored type of dimensionality reduction is sparsity; another type is generative modelling of signal distributions. Generative models based on neural networks, such as GANs or variational auto-encoders, are particularly performant and are gaining on applicability. In this paper we study spiked matrix models, where a low-rank matrix is observed through a noisy channel. This problem with sparse structure of the spikes has attracted broad attention in the past literature. Here, we replace the sparsity assumption by generative modelling, and investigate the consequences on statistical and algorithmic properties. We analyze the Bayes-optimal performance under specific generative models for the spike. In contrast with the sparsity assumption, we do not observe regions of parameters where statistical performance is superior to the best known algorithmic performance. We show that in the analyzed cases the approximate message passing algorithm is able to reach optimal performance. We also design enhanced spectral algorithms and analyze their performance and thresholds using random matrix theory, showing their superiority to the classical principal component analysis. We complement our theoretical results by illustrating the performance of the spectral algorithms when the spikes come from real datasets.