Publications by Nicolas Macris

Rectangular rotational invariant estimator for high-rank matrix estimation

We consider estimating a matrix from noisy observations coming from an arbitrary additive bi-rotational invariant perturbation. We propose an estimator, which we conjecture is optimal among the class of rectangular rotational invariant estimators and can b ...

Oxford University Press (OUP)2025

On the Convergence to the Non-equilibrium Steady State of a Langevin Dynamics with Widely Separated Time Scales and Different Temperatures

Nicolas Macris, Emanuele Mingione, Diego Alberici

We study the solution of the two-temperature Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are e ...

Springer Int Publ Ag2024

Matrix Inference in Growing Rank Regimes

Nicolas Macris, Farzad Pourkamali

The inference of a large symmetric signal-matrix S ϵ RN × N corrupted by additive Gaussian noise, is considered for two regimes of growth of the rank M as a function of N. For sub-linear ranks M=Θ (Nα) with α in (0,1) the mutual information and minimum mea ...

2024

Solving Non-linear Kolmogorov Equations in Large Dimensions by Using Deep Learning: A Numerical Comparison of Discretization Schemes

Nicolas Macris

Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation describes pattern f ...

SPRINGER/PLENUM PUBLISHERS2023

Statistical limits of dictionary learning: Random matrix theory and the spectral replica method

Nicolas Macris, Jean François Emmanuel Barbier

We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existin ...

AMER PHYSICAL SOC2022

Mismatched Estimation of Non-Symmetric Rank-One Matrices Under Gaussian Noise

Nicolas Macris, Farzad Pourkamali

We consider the estimation of a nxm matrix uv(T) observed through an additive Gaussian noise channel, a problem that frequently arises in statistics and machine learning. We investigate a scenario involving mismatched Bayesian inference in which the stat ...

IEEE2022

Information theoretic limits of learning a sparse rule

Nicolas Macris, Clément Dominique Luneau

We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per ...

IOP Publishing Ltd2022

Adaptive Path Interpolation Method for Sparse Systems: Application to a Censored Block Model

Nicolas Macris, Chun Lam Chan, Jean François Emmanuel Barbier

Recently, a new adaptive path interpolation method has been developed as a simple and versatile scheme to calculate exactly the asymptotic mutual information of Bayesian inference problems defined on dense factor graphs. These include random linear and gen ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021

Bell Diagonal and Werner State Generation: Entanglement, Non-Locality, Steering and Discord on the IBM Quantum Computer

Marc-André Dupertuis, Nicolas Macris, Nicolas Schwaller, Chun Lam Chan, Frédéric Gessler, Su Yeon Chang, Samuel Bosch

We propose the first correct special-purpose quantum circuits for preparation of Bell diagonal states (BDS), and implement them on the IBM Quantum computer, characterizing and testing complex aspects of their quantum correlations in the full parameter spac ...

2021

Mutual information for low-rank even-order symmetric tensor estimation

Nicolas Macris, Jean François Emmanuel Barbier, Clément Dominique Luneau

We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method orig ...

OXFORD UNIV PRESS2021

Tensor estimation with structured priors

Nicolas Macris, Clément Dominique Luneau

We consider rank-one symmetric tensor estimation when the tensor is corrupted by gaussian noise and the spike forming the tensor is a structured signal coming from a generalized linear model. The latter is a mathematically tractable model of a non-trivial ...

2020

High-dimensional rank-one nonsymmetric matrix decomposition: the spherical case

Nicolas Macris, Jean François Emmanuel Barbier, Clément Dominique Luneau

We consider the problem of estimating a rank-one nonsymmetric matrix under additive white Gaussian noise. The matrix to estimate can be written as the outer product of two vectors and we look at the special case in which both vectors are uniformly distribu ...

IEEE2020

Concentration of Multi-overlaps for Random Dilute Ferromagnetic Spin Models

Nicolas Macris, Chun Lam Chan, Jean François Emmanuel Barbier

We consider mean field ferromagnetic spin models on dilute random graphs and prove that, with suitable one-body infinitesimal perturbations added to the Hamiltonian, the multi-overlaps concentrate for all temperatures, both with respect to the thermal Gibb ...

SPRINGER2020

Mutual Information and Optimality of Approximate Message-Passing in Random Linear Estimation

Nicolas Macris, Mohamad Baker Dia, Jean François Emmanuel Barbier, Florent Gérard Krzakala

We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections. A few examples where this problem is relevant are compressed sensing, sparse superposition codes, and code division multiple access. There has been a ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2020

The committee machine: computational to statistical gaps in learning a two-layers neural network

Nicolas Macris, Jean François Emmanuel Barbier, Florent Gérard Krzakala, Lenka Zdeborová

Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks. In this paper, we provide a rigor ...

IOP PUBLISHING LTD2019

Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

Nicolas Macris, Chun Lam Chan, Jean François Emmanuel Barbier

We rigorously derive a single-letter variational expression for the mutual information of the asymmetric two-groups stochastic block model in the dense graph regime. Existing proofs in the literature are indirect, as they involve mapping the model to a ran ...

IEEE2019

The adaptive interpolation method: a simple scheme to prove replica formulas in Bayesian inference

Nicolas Macris, Jean François Emmanuel Barbier

In recent years important progress has been achieved towards proving the validity of the replica predictions for the (asymptotic) mutual information (or free energy) in Bayesian inference problems. The proof techniques that have emerged appear to be quite ...

SPRINGER HEIDELBERG2019

The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models

Nicolas Macris, Jean François Emmanuel Barbier

In this contribution we give a pedagogic introduction to the newly introduced adaptive interpolation method to prove in a simple and unified way replica formulas for Bayesian optimal inference problems. Many aspects of this method can already be explained ...

IOP PUBLISHING LTD2019

Displacement Convexity in Spatially Coupled Scalar Recursions

Nicolas Macris, Rüdiger Urbanke, Rafah El-Khatib

We introduce a technique for the analysis of general spatially coupled systems that are governed by scalar recursions. Such systems can be expressed in variational form in terms of a potential function. We show, under mild conditions, that the potential fu ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2019

Entropy and mutual information in models of deep neural networks

Nicolas Macris, Jean François Emmanuel Barbier, Clément Dominique Luneau, Florent Gérard Krzakala, Lenka Zdeborová

We examine a class of stochastic deep learning models with a tractable method to compute information-theoretic quantities. Our contributions are three-fold: (i) we show how entropies and mutual informations can be derived from heuristic statistical physics ...

2019