# Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models

arXiv: https://arxiv.org/abs/2403.03695 · HTML: https://mlsanity.com/paper/2403.03695/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 65 references that are in the map.
- First posted on arXiv 6 Mar 2024.

## Builds on (most cited first, 17 of 17)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Fundamental Limits of Weak Recovery with Applications to Phase Retrieval](https://mlsanity.com/paper/1708.05932/) (2017)
- [The estimation error of general first order methods](https://mlsanity.com/paper/2002.12903/) (2020)
- [The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics](https://mlsanity.com/paper/2205.09727/) (2022)
- [Spectral Clustering of Graphs with the Bethe Hessian](https://mlsanity.com/paper/1406.1880/) (2014)
- [Notes on computational-to-statistical gaps: predictions using statistical physics](https://mlsanity.com/paper/1803.11132/) (2018)
- [Approximate Message Passing with Spectral Initialization for Generalized Linear Models](https://mlsanity.com/paper/2010.03460/) (2020)
- [The spiked matrix model with generative priors](https://mlsanity.com/paper/1905.12385/) (2019)
- [Fundamental limits of detection in the spiked Wigner model](https://mlsanity.com/paper/1806.09588/) (2018)
- [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021)
- [Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models](https://mlsanity.com/paper/2008.03326/) (2020)
- [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 limits of a multiview low-rank symmetric spiked matrix model](https://mlsanity.com/paper/2005.08017/) (2020)
- [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)
- [Approximate Message Passing for the Matrix Tensor Product Model](https://mlsanity.com/paper/2306.15580/) (2023)

## Built on by (most cited first, 4 of 4)

- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions](https://mlsanity.com/paper/2407.03522/) (2024)
- [Sparse corruption in low-rank matrix inference: the PCA benchmark](https://mlsanity.com/paper/2511.11927/) (2025)
- [Implicitly Normalized Online PCA: A Regularized Algorithm with Exact High-Dimensional Dynamics](https://mlsanity.com/paper/2512.01231/) (2025)

## Abstract (arXiv)

We discuss the inhomogeneous spiked Wigner model, a theoretical framework recently introduced to study structured noise in various learning scenarios, through the prism of random matrix theory, with a specific focus on its spectral properties. Our primary objective is to find an optimal spectral method and to extend the celebrated citeBBP (BBP) phase transition criterion -- well-known in the homogeneous case -- to our inhomogeneous, block-structured, Wigner model. We provide a thorough rigorous analysis of a transformed matrix and show that the transition for the appearance of 1) an outlier outside the bulk of the limiting spectral distribution and 2) a positive overlap between the associated eigenvector and the signal, occurs precisely at the optimal threshold, making the proposed spectral method optimal within the class of iterative methods for the inhomogeneous Wigner problem.