Estimation

The Mutual Information in Random Linear Estimation Beyond i.i.d. Matrices

There has been definite progress recently in proving the variational single-letter formula given by the heuristic replica method for various estimation problems. In particular, the replica formula for the mutual information in the case of noisy …

Multi-layer generalized linear estimation

We consider the problem of reconstructing a signal from multi-layered (possibly) non-linear measurements. Using non-rigorous but standard methods from statistical physics we present the Multi-Layer Approximate Message Passing (ML-AMP) algorithm for …

Statistical and computational phase transitions in spiked tensor estimation

We consider tensor factorizations using a generative model and a Bayesian approach. We compute rigorously the mutual information, the Minimal Mean Square Error (MMSE), and unveil information-theoretic phase transitions. In addition, we study the …

Performance Limits for Noisy Multimeasurement Vector Problems

Compressed sensing (CS) demonstrates that sparse signals can be estimated from underdetermined linear systems. Distributed CS (DCS) further reduces the number of measurements by considering joint sparsity within signal ensembles. DCS with jointly …

Mutual information in rank-one matrix estimation

We consider the estimation of a n-dimensional vector x from the knowledge of noisy and possibility non-linear element-wise measurements of xxT, a very generic problem that contains, e.g. stochastic 2-block model, submatrix localization or the spike …

Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering

We consider the problem of Gaussian mixture clustering in the high-dimensional limit where the data consists of m points in n dimensions, n,m → ∞ and α = m/n stays finite. Using exact but non-rigorous methods from statistical physics, we determine …

The mutual information in random linear estimation

We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections, a problem relevant in compressed sensing, sparse superposition codes or code division multiple access just to cite few. There has been a number …

MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel

This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but …

Phase transitions in sparse PCA

We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state evolution to …

Phase recovery from a Bayesian point of view: The variational approach

In this paper, we consider the phase recovery problem, where a complex signal vector has to be estimated from the knowledge of the modulus of its linear projections, from a naive variational Bayesian point of view. In particular, we derive an …