Publications de Nicolas Macris

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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 ...

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 ...

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 ...

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 ...

Generalized linear models (GLMs) are used 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-corre ...

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 ...

We consider a gauge symmetric version of the p-spin glass model on a complete graph. The gauge symmetry guarantees the absence of replica symmetry breaking and allows to fully use the interpolation scheme of Guerra to rigorously compute the free energy. In ...

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 contribution, we provide ...

A new adaptive path interpolation method has been recently 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 gene ...

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 ...

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 ...

Sparse superposition codes, or sparse regression codes, constitute a new class of codes, which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the A ...

We analyze the time reversed dynamics of generative diffusion models. If the exact empirical score function is used in a regime of large dimension and exponentially large number of samples, these models are known to undergo transitions between distinct dynamical regimes. We extend this analysis and compute the transitions for an analytically tractable manifold model where the statistical model for the data is a mixture of lower dimensional Gaussians embedded in higher dimensional space. We compute the so-ca

The Falicov–Kimball model consists of itinerant lattice fermions interacting with Ising spins by an on-site potential of strength U. Kennedy and Lieb proved that at half filling there is a low temperature phase with chessboard long range order on Zd, d \ 2, for all non-zero values of U. Here we investigate the stability of this phase when small quantum fluctuations of the ‘‘Ising spins’’ are introduced in two different ways. The first one corresponds to replace the classical spins by quantum two level syste

We derive various sum rules for the time-displaced structure function of a classical one-component plasma subjected to an external uniform magnetic field. When the plasma has some translational invariance (i.e., homogeneous or translation-invariant along the field), we find that there are long-wavelength oscillations with well-defined frequencies. The results are obtained from linear response and macroscopic electrodynamics, as well as from the microscopic equations of motion (BBGKY hierarchy). In the prese

We consider a collection of Curie-Weiss (CW) spin systems, possibly with a random field, each of which is placed along the positions of a one-dimensional chain. The CW systems are coupled together by a Kac-type interaction in the longitudinal direction of ...

We consider communication over a binary erasure channel with low density parity check codes and optimal maximum a posteriori decoding. It is known that the problem of computing the average conditional entropy, over such code ensembles, in the asymptotic li ...

Using Fefferman's analysis of the quantum electron-proton gas, we give a rigorous proof of ionization equilibrium in this system. Ionization equilibrium phases are obtained as low-density and low-temperature limits, letting the chemical potentialmgr(T) approach the ground-state energy of the hydrogen atom as the temperatureT tends to zero. The rate of ionization is determined by the slope ofmgr(T) atT=0 and is correctly given by the Saha formula. We also discuss a simpler model where a single quantum partic

We review, from a unified point of view, a general class of models of itinerant electrons interacting with classical fields. Applications to the static Holstein, Kondo, and Hubbard models are discussed. The ground state structure of the classical field is investigated when the electron band is half-filled. Some of the results are also valid when there is a Hubbard interaction between spin up and spin down electrons. It is found that the ground states are either homogeneous or period two Ne´el configurations

Nicolas Macris

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

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

Displacement Convexity in Spatially Coupled Scalar Recursions

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

Optimal errors and phase transitions in high-dimensional generalized linear models

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

Exact Solution of the Gauge Symmetric p-Spin Glass Model on a Complete Graph

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

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

Entropy and mutual information in models of deep neural networks

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

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Analysis of Diffusion Models for Manifold Data

Long range orders in models of itinerant electrons interacting with heavy quantum fields

Time dependent correlations of a one component plasma in a uniform magnetic field

Chains of mean-field models

Exact solution for the conditional entropy of Poissonian LDPC codes over the Binary Erasure Channel

Ionisation equilibrium in the electron-proton gas

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

Ground states and low temperature phases of itinerant electrons interacting with classical fields: a review of rigorous results