# All-or-nothing statistical and computational phase transitions in sparse spiked matrix estimation

We determine statistical and computational limits for estimation of a rank-one matrix (the spike) corrupted by an additive gaussian noise matrix, in a sparse limit, where the underlying hidden vector (that constructs the rank-one matrix) has a number of non-zero components that scales sub-linearly with the total dimension of the vector, and the signal-to-noise ratio tends to infinity at an appropriate speed. We prove explicit low-dimensional variational formulas for the asymptotic mutual information between the spike and the observed noisy matrix and analyze the approximate message passing algorithm in the sparse regime. For Bernoulli and Bernoulli-Rademacher distributed vectors, and when the sparsity and signal strength satisfy an appropriate scaling relation, we find all-or-nothing phase transitions for the asymptotic minimum and algorithmic mean-square errors. These jump from their maximum possible value to zero, at well defined signal-to-noise thresholds whose asymptotic values we determine exactly. In the asymptotic regime the statistical-to-algorithmic gap diverges indicating that sparse recovery is hard for approximate message passing.

## Authors

• 22 publications
• 20 publications
• 20 publications
11/12/2019

### 0-1 phase transitions in sparse spiked matrix estimation

We consider statistical models of estimation of a rank-one matrix (the s...
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### The fundamental limits of sparse linear regression with sublinear sparsity

We establish exact asymptotic expressions for the normalized mutual info...
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### The spiked matrix model with generative priors

Using a low-dimensional parametrization of signals is a generic and powe...
03/01/2015

### Phase Transitions in Sparse PCA

We study optimal estimation for sparse principal component analysis when...
05/25/2021

### Rank-one matrix estimation: analytic time evolution of gradient descent dynamics

We consider a rank-one symmetric matrix corrupted by additive noise. The...
06/22/2021

### Rank-one matrix estimation with groupwise heteroskedasticity

We study the problem of estimating a rank-one matrix from Gaussian obser...
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