# Information-Theoretic Bounds and Phase Transitions in Clustering, Sparse PCA, and Submatrix Localization

@article{Banks2018InformationTheoreticBA, title={Information-Theoretic Bounds and Phase Transitions in Clustering, Sparse PCA, and Submatrix Localization}, author={Jessica E. Banks and Cristopher Moore and Roman Vershynin and Nicolas Verzelen and Jiaming Xu}, journal={IEEE Transactions on Information Theory}, year={2018}, volume={64}, pages={4872-4894} }

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these…

## 64 Citations

### Fundamental limits of symmetric low-rank matrix estimation

- Computer ScienceCOLT
- 2017

This paper considers the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise and compute the limit in the large dimension setting for the mutual information between the signal and the observations, while the rank of the signal remains constant.

### Phase transitions in spiked matrix estimation: information-theoretic analysis

- Computer ScienceArXiv
- 2018

The minimal mean squared error is computed for the estimation of the low-rank signal and it is compared to the performance of spectral estimators and message passing algorithms.

### Rank-one matrix estimation with groupwise heteroskedasticity

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- 2021

This work proves asymptotically exact formulas for the minimum mean-squared error in estimating both the matrix and the latent variables of a rank-one matrix from Gaussian observations.

### Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization

- Computer ScienceArXiv
- 2016

The fundamental limitations of statistical methods are studied, including non-spectral ones, and it is shown that inefficient procedures can work below the threshold where PCA succeeds, whereas no known efficient algorithm achieves this.

### Statistical limits of spiked tensor models

- Computer ScienceAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2020

The replica prediction from statistical physics is conjectured to give the exact information-theoretic threshold for any fixed $d$, and a new improvement to the second moment method for contiguity is introduced, on which the lower bounds are based.

### Reducibility and Computational Lower Bounds for Problems with Planted Sparse Structure

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- 2018

This work introduces several new techniques to give a web of average-case reductions showing strong computational lower bounds based on the planted clique conjecture using natural problems as intermediates, including tight lower bounds for Planted Independent Set, Planted Dense Subgraph, Sparse Spiked Wigner, and Sparse PCA.

### Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models

- Computer ScienceThe Annals of Statistics
- 2018

The statistical limits of tests for the presence of a spike are studied, including non-spectral tests, and include the Gaussian Wigner ensemble, where it is shown that PCA achieves the optimal detection threshold for certain natural priors for the spike.

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### Finite Size Corrections and Likelihood Ratio Fluctuations in the Spiked Wigner Model

- Computer Science, MathematicsArXiv
- 2017

It is proved that below the reconstruction threshold, where it becomes impossible to reconstruct the spike, the log-likelihood ratio has fluctuations of constant order and converges in distribution to a Gaussian under both the planted and (under restrictions) the null model.

### Fundamental limits of low-rank matrix estimation: the non-symmetric case

- Computer Science
- 2017

This work considers the high-dimensional inference problem where the signal is a low-rank matrix which is corrupted by an additive Gaussian noise and compute the limit in the large dimension setting for the mutual information between the signal and the observations, as well as the matrix minimum mean square error.

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