matrix algebra

Estimation in the Spiked Wigner Model: A Short Proof of the Replica Formula

We consider the problem of estimating the rank-one perturbation of a Wigner matrix in a setting of low signal-to-noise ratio. This serves as a simple model for principal component analysis in high dimensions. The mutual information per variable …

Scaling Up Echo-State Networks With Multiple Light Scattering

Echo-State Networks and Reservoir Computing have been studied for more than a decade. They provide a simpler yet powerful alternative to Recurrent Neural Networks, every internal weight is fixed and only the last linear layer is trained. They involve …

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 …

Streaming Bayesian inference: Theoretical limits and mini-batch approximate message-passing

In statistical learning for real-world large-scale data problems, one must often resort to “streaming” algorithms which operate sequentially on small batches of data. In this work, we present an analysis of the information-theoretic limits of …

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 …

Fast Phase Retrieval for High Dimensions: A Block-Based Approach

This paper addresses fundamental scaling issues that hinder phase retrieval (PR) in high dimensions. We show that, if the measurement matrix can be put into a generalized block-diagonal form, a large PR problem can be solved on separate blocks, at …

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 …

Intensity-only optical compressive imaging using a multiply scattering material and a double phase retrieval approach

In this paper, the problem of compressive imaging is addressed using natural randomization by means of a multiply scattering medium. To utilize the medium in this way, its corresponding transmission matrix must be estimated. For calibration purposes, …

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 …

Spectral detection on sparse hypergraphs

We consider the problem of the assignment of nodes into communities from a set of hyperedges, where every hyperedge is a noisy observation of the community assignment of the adjacent nodes. We focus in particular on the sparse regime where the number …