Publications by Nicolas Macris

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

Jean François Emmanuel Barbier, Nicolas Macris

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 existing literature concerned with the low-rank (i.e., constant-rank) regime. We first consider a class of rotationally invariant matrix denoising problems whose mutual information and minimum mean-square error are computable using techniques from random matrix theory. Next, we analyze the more challenging models of dictionary learning. To do so we introduce a combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. This allows us to derive variational formulas for the mutual information between hidden representations and the noisy data of the dictionary learning problem, as well as for the overlaps quantifying the optimal reconstruction error. The proposed method reduces the number of degrees of freedom from circle minus(N-2) matrix entries to circle minus(N) eigenvalues (or singular values), and yields Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are a combination of large deviation results for random matrices together with a replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices and the use of Harish-Chandra-Itzykson-Zuber spherical integrals.

AMER PHYSICAL SOC2022

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

Jean François Emmanuel Barbier, Chun Lam Chan, Nicolas Macris

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 generalized estimation, sparse superposition codes, and low-rank matrix / tensor estimation. For all these systems, the adaptive interpolation method directly proves that the replica-symmetric prediction is exact, in a simple and unified manner. When the underlying factor graph of the inference problem is sparse the replica prediction is considerably more complicated, and rigorous results are often lacking or obtained by rather complicated methods. In this work we show how to extend the adaptive path interpolation method to sparse systems. We concentrate on a censored block model, where hidden variables are measured through a binary erasure channel, for which we fully prove the replica prediction.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021

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

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

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 space. Among the circuits proposed, one involves only two quantum bits but requires adapted quantum tomography routines handling classical bits in parallel. The entire class of Bell diagonal states is generated, and several characteristic indicators, namely entanglement of formation and concurrence, CHSH non-locality, steering and discord, are experimentally evaluated over the full parameter space and compared with theory. As a by-product of this work, we also find a remarkable general inequality between “quantum discord” and “asymmetric relative entropy of discord”: the former never exceeds the latter. We also prove that for all BDS the two coincide.

2021

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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 originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new non-trivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.

OXFORD UNIV PRESS2021

Concentration of Multi-overlaps for Random Dilute Ferromagnetic Spin Models

Jean François Emmanuel Barbier, Chun Lam Chan, Nicolas Macris

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 Gibbs average and the quenched randomness. Results of this nature have been known only for the lowest order overlaps, at high temperature or on the Nishimori line. Here we treat all multi-overlaps by a non-trivial application of Griffiths-Kelly-Sherman correlation inequalities. Our results apply in particular to the pure and mixedp-spin ferromagnets on random dilute Erdoes-Renyi hypergraphs. On physical grounds one expects that multi-overlap concentration is an important ingredient for the validity of the cavity (or replica-symmetric) formula for the pressure of mean field models. However rigorously establishing this formula for thep-spin ferromagnet on a random dilute hypergraph remains an open problem.

SPRINGER2020

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

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

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 number of works considering the mutual information for this problem using the replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-Toninelli type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical cases, we present a converse lower bound via a method that uses spatial coupling, state evolution analysis and the I-MMSE theorem. This yields a single letter formula for the mutual information and the minimal-mean-square error for random Gaussian linear estimation of all discrete bounded signals. In addition, we prove that the low complexity approximate message-passing algorithm is optimal outside of the so-called hard phase, in the sense that it asymptotically reaches the minimal-mean-square error. In this work spatial coupling is used primarily as a proof technique. However our results also prove two important features of spatially coupled noisy linear random Gaussian estimation. First there is no algorithmically hard phase. This means that for such systems approximate message-passing always reaches the minimal-mean-square error. Secondly, in the limit of infinitely long coupled chain, the mutual information associated to spatially coupled systems is the same as the one of uncoupled linear random Gaussian estimation.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2020

Displacement Convexity in Spatially Coupled Scalar Recursions

Rafah El-Khatib, Nicolas Macris, Rüdiger Urbanke

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 function is displacement convex and that the minimizers are given by the fixed points (FPs) of the recursions. Furthermore, we give the conditions on the system such that the minimizing FP is unique up to translation along the spatial direction. The condition matches with that of Kudekar et al. [20] for the existence of spatial FPs. Displacement convexity applies to a wide range of spatially coupled recursions appearing in coding theory, compressive sensing, random constraint satisfaction problems, as well as statistical-mechanics models. We illustrate it with applications to low-density parity-check (LDPC) and generalized LDPC codes used for the transmission on the binary erasure channel or general binary memoryless symmetric channels within the Gaussian reciprocal channel approximation as well as compressive sensing.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2019

Efficient Quantum Algorithms for GHZ and W States, and Implementation on the IBM Quantum Computer

Chun Lam Chan, Marc-André Dupertuis, Romain Fournier, Fabien Gremion, Clément Christian Javerzac-Galy, Kenichi Komagata, Nicolas Macris, Jarla Thiesbrummel

Efficient deterministic algorithms are proposed with logarithmic step complexities for the generation of entangled GHZ(N) and W-N states useful for quantum networks, and an implementation on the IBM quantum computer up to N = 16 is demonstrated. Improved quality is then investigated using full quantum tomography for low-N GHZ and W states. This is completed by parity oscillations and histogram distance for large-N GHZ and W states, respectively. Robust states are built with about twice the number of quantum bits which were previously achieved.

WILEY2019

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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 methods, under the assumption that weight matrices are independent and orthogonally-invariant. (ii) We extend particular cases in which this result is known to be rigorously exact by providing a proof for two-layers networks with Gaussian random weights, using the recently introduced adaptive interpolation method. (iii) We propose an experiment framework with generative models of synthetic datasets, on which we train deep neural networks with a weight constraint designed so that the assumption in (i) is verified during learning. We study the behavior of entropies and mutual informations throughout learning and conclude that, in the proposed setting, the relationship between compression and generalization remains elusive.

IOP PUBLISHING LTD2019

Mutual Information for Low-Rank Even-Order Symmetric Tensor Factorization

Jean François Emmanuel Barbier, Clément Dominique Luneau, Nicolas Macris

We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its mutual information when the tensor is of even order. The proof uses the adaptive interpolation method, for which rank-one matrix factorization is one of the first problems to which it was successfully applied. We show how to extend the adaptive interpolation to finite-rank symmetric tensors of even order, which requires new ideas with respect to the proof for the rank-one case. We also underline where the proof falls short when dealing with odd-order tensors.

IEEE2019