# Sparse corruption in low-rank matrix inference: the PCA benchmark

arXiv: https://arxiv.org/abs/2511.11927 · HTML: https://mlsanity.com/paper/2511.11927/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 82 references that are in the map.
- First posted on arXiv 14 Nov 2025.

## Builds on (most cited first, 14 of 14)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Spectral redemption: clustering sparse networks](https://mlsanity.com/paper/1306.5550/) (2013)
- [SLTrain: a sparse plus low-rank approach for parameter and memory efficient pretraining](https://mlsanity.com/paper/2406.02214/) (2024)
- [Non-convex Robust PCA](https://mlsanity.com/paper/1410.7660/) (2014)
- [Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs](https://mlsanity.com/paper/1501.06087/) (2015)
- [Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula](https://mlsanity.com/paper/1606.04142/) (2016)
- [MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel](https://mlsanity.com/paper/1507.03857/) (2015)
- [OATS: Outlier-Aware Pruning Through Sparse and Low Rank Decomposition](https://mlsanity.com/paper/2409.13652/) (2024)
- [The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation?](https://mlsanity.com/paper/2205.10009/) (2022)
- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models](https://mlsanity.com/paper/2403.03695/) (2024)
- [Spectral density of the non-backtracking operator](https://mlsanity.com/paper/1404.7787/) (2014)
- [Robust PCA for High Dimensional Data based on Characteristic Transformation](https://mlsanity.com/paper/2204.01042/) (2022)
- [Fundamental limits of Non-Linear Low-Rank Matrix Estimation](https://mlsanity.com/paper/2403.04234/) (2024)

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

- [Spectral phase transitions and trainability in neural network learning dynamics](https://mlsanity.com/paper/2606.28486/) (2026)
- [Rank-One Signal Recovery in Sparse Wishart Noise](https://mlsanity.com/paper/2609.28163/) (2026)

## Abstract (arXiv)

Principal Component Analysis (PCA) is a standard tool for extracting a low-rank signal from noisy observations. It is known that applying PCA to a rank-one signal corrupted by a dense, homogeneous noise, in the large matrix size limit, the celebrated BBP transition occurs, where the emergence of an outlying eigenvalue and the alignment of the corresponding eigenvector occur at the same critical signal strength. Here we study the case of sparse noise corruption. The noise matrix is modelled as the adjacency matrix of a weighted undirected graph with finite average connectivity. Using the replica method, we analytically compute the typical top eigenvalue, the top eigenvector component density, and the squared overlap with the signal, through recursive distributional equations solved by population dynamics. We identify two signal-strength transitions as functions of graph connectivity: θrm crit, marking signal recovery by the top eigenvector and generalising the BBP transition, and θrm b, where the signal-related eigenvalue detaches from the bulk. For noise with nonzero mean, these transitions need not coincide because of a structural sparse-graph outlier, leading to a discontinuous transition in the squared overlap with the top eigenvector. The same top-eigenpair formalism also predicts the overlap of the signal with the eigenvector associated with the second largest eigenvalue when the signal eigenvalue is an outlier but remains below the structural outlier, where the transition is continuous. We specialise the equations to Poissonian and Random Regular degree distributions, recover dense-noise results in the large-connectivity limit, and validate the theory by numerical diagonalisation of large matrices.