phase transitions

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 …

Decoding from pooled data: Phase transitions of message passing

We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an Approximate Message Passing (AMP) algorithm for recovering the signal in the random dense …

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 …

Phase Transitions and Sample Complexity in Bayes-Optimal Matrix Factorization

We analyze the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications, such as dictionary learning, blind matrix calibration, sparse …

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 …

Compressed sensing of approximately-sparse signals: Phase transitions and optimal reconstruction

Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only “approximately sparse”, i.e. even though the signal contains only a small fraction of relevant (large) components the …

Gibbs states and the set of solutions of random constraint satisfaction problems

An instance of a random constraint satisfaction problem defines a random subset 𝒮 (the set of solutions) of a large product space X N (the set of assignments). We consider two prototypical problem ensembles (random k-satisfiability and q-coloring of …