Generalized linear models (GLMs) arise in high-dimensional machine learning, statistics, communications and signal processing. % In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing, error-correcting codes or benchmarks models in neural networks. % We evaluate the mutual information (or “free entropy”) from which we deduce the Bayes-optimal inference and generalization errors. Our analysis applies to the high-dimensional limit where both the number of samples and dimensions are large and their ratio is fixed. % Non-rigorous predictions for the optimal inference and generalization errors existed for special cases of GLMs, e.g. for the perceptron in the field of statistical physics based on the so-called replica method. Our present paper rigorously establishes those decades old conjectures and brings forward their algorithmic interpretation in terms of performance of the generalized approximate message-passing algorithm. % Furthermore, we tightly characterize, for many learning problems, regions of parameters for which this algorithm achieves the optimal performance, and locate the associated sharp phase transitions separating learnable and non-learnable regions.

We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we 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, while the rank of the signal remains constant. We unify and generalize a number of recent works on PCA, sparse PCA, submatrix localization or community detection by computing the information-theoretic limits for these problems in the high noise regime. This allows to locate precisely the information-theoretic thresholds for the above mentioned problems.

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 of Approximate Message Passing (AMP) and show that it achieves the MMSE for a large set of parameters, and that factorization is algorithmically “easy” in a much wider region than previously believed. It exists, however, a “hard” region where AMP fails to reach the MMSE and we conjecture that no polynomial algorithm will improve on AMP.