# Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity

arXiv: https://arxiv.org/abs/2106.11950 · HTML: https://mlsanity.com/paper/2106.11950/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 56 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 20 of 20)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Contextual Stochastic Block Models](https://mlsanity.com/paper/1807.09596/) (2018)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Estimation of Low-Rank Matrices via Approximate Message Passing](https://mlsanity.com/paper/1711.01682/) (2017)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models](https://mlsanity.com/paper/1807.00891/) (2018)
- [Covariate-assisted spectral clustering](https://mlsanity.com/paper/1411.2158/) (2014)
- [Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula](https://mlsanity.com/paper/1606.04142/) (2016)
- [Sparse PCA via Covariance Thresholding](https://mlsanity.com/paper/1311.5179/) (2013)
- [The spiked matrix model with generative priors](https://mlsanity.com/paper/1905.12385/) (2019)
- [All-or-nothing statistical and computational phase transitions in sparse spiked matrix estimation](https://mlsanity.com/paper/2006.07971/) (2020)
- [Phase Transitions in Sparse PCA](https://mlsanity.com/paper/1503.00338/) (2015)
- [Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering](https://mlsanity.com/paper/1610.02918/) (2016)
- [Rank-one matrix estimation: analysis of algorithmic and information theoretic limits by the spatial coupling method](https://mlsanity.com/paper/1812.02537/) (2018)
- [How Many Samples is a Good Initial Point Worth in Low-rank Matrix Recovery?](https://mlsanity.com/paper/2006.06915/) (2020)
- [Information-Theoretic Limits for the Matrix Tensor Product](https://mlsanity.com/paper/2005.11273/) (2020)
- [Information-theoretic limits of a multiview low-rank symmetric spiked matrix model](https://mlsanity.com/paper/2005.08017/) (2020)
- [Mutual Information in Community Detection with Covariate Information and Correlated Networks](https://mlsanity.com/paper/1912.05375/) (2019)

## Built on by (most cited first, 8 of 8)

- [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)
- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models](https://mlsanity.com/paper/2403.03695/) (2024)
- [Dimension-Free Bounds for Generalized First-Order Methods via Gaussian Coupling](https://mlsanity.com/paper/2508.10782/) (2025)
- [Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noise](https://mlsanity.com/paper/2510.08541/) (2025)
- [Precise sample covariance spectral norm error -- an RDT view](https://mlsanity.com/paper/2607.14460/) (2026)
- [Heteroskedastic Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/2610.00498/) (2026)

## Abstract (arXiv)

Low-rank matrix recovery problems involving high-dimensional and heterogeneous data appear in applications throughout statistics and machine learning. The contribution of this paper is to establish the fundamental limits of recovery for a broad class of these problems. In particular, we study the problem of estimating a rank-one matrix from Gaussian observations where different blocks of the matrix are observed under different noise levels. In the setting where the number of blocks is fixed while the number of variables tends to infinity, we prove asymptotically exact formulas for the minimum mean-squared error in estimating both the matrix and underlying factors. These results are based on a novel reduction from the low-rank matrix tensor product model (with homogeneous noise) to a rank-one model with heteroskedastic noise. As an application of our main result, we show that recently proposed methods based on applying principal component analysis (PCA) to weighted combinations of the data are optimal in some settings but sub-optimal in others. We also provide numerical results comparing our asymptotic formulas with the performance of methods based on weighted PCA, gradient descent, and approximate message passing.