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Rémi Monasson

Rémi Monasson appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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cond-mat.stat-mechcond-mat.dis-nnBiological PhysicsNeurons and CognitionQuantitative MethodsBiomoleculesComputational ComplexityMachine LearningMethodologyphysics.data-an

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Published work

17 published item(s)

preprint2026arXiv

Spherical Boltzmann machines: a solvable theory of learning and generation in energy-based models

Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics, but their learning and generative properties remain poorly understood. Here, we analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM). Combining tools from random matrix theory and dynamical mean-field theory, we: solve exact equations describing the training dynamics of the SBM; compute the Bayesian evidence, which acts as a partition function in parameter space and encodes global properties of the trained model; and uncover cascades of phase transitions that occur both during training and as a function of hyperparameters, related to successive alignment and condensation of the top modes of the coupling matrix to the data. We connect these transitions to sampling-time generative phenomena in a teacher-student scenario, including: sampling temperature tuning, double descent as a function of regularization strength, tempered posterior effects, and out-of-equilibrium effects during training that induce biases in the trained model. We provide numerical evidence demonstrating that all these phenomena appear in standard generative architectures, beyond the SBM.

0 citations0 reviewsNo code linkedOpen access
preprint2024arXiv

Information content in continuous attractor neural networks is preserved in the presence of moderate disordered background connectivity

Continuous attractor neural networks (CANN) form an appealing conceptual model for the storage of information in the brain. However a drawback of CANN is that they require finely tuned interactions. We here study the effect of quenched noise in the interactions on the coding of positional information within CANN. Using the replica method we compute the Fisher information for a network with position-dependent input and recurrent connections composed of a short-range (in space) and a disordered component. We find that the loss in positional information is small for not too large disorder strength, indicating that CANN have a regime in which the advantageous effects of local connectivity on information storage outweigh the detrimental ones. Furthermore, a substantial part of this information can be extracted with a simple linear readout.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks

Recurrent neural networks (RNN) are powerful tools to explain how attractors may emerge from noisy, high-dimensional dynamics. We study here how to learn the ~N^(2) pairwise interactions in a RNN with N neurons to embed L manifolds of dimension D << N. We show that the capacity, i.e. the maximal ratio L/N, decreases as |log(epsilon)|^(-D), where epsilon is the error on the position encoded by the neural activity along each manifold. Hence, RNN are flexible memory devices capable of storing a large number of manifolds at high spatial resolution. Our results rely on a combination of analytical tools from statistical mechanics and random matrix theory, extending Gardner's classical theory of learning to the case of patterns with strong spatial correlations.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Gaussian Closure Scheme in the Quasi-Linkage Equilibrium Regime of Evolving Genome Populations

Describing the evolution of a population of genomes evolving in a complex fitness landscape is generally very hard. We here introduce an approximate Gaussian closure scheme to characterize analytically the statistics of a genomic population in the so-called Quasi--Linkage Equilibrium (QLE) regime, applicable to generic values of the rates of mutation or recombination and fitness functions. The Gaussian approximation is illustrated on a short-range fitness landscape with two far away and competing maxima. It unveils the existence of a phase transition from a broad to a polarized distribution of genomes as the strength of epistatic couplings is increased, characterized by slow coarsening dynamics of competing allele domains. Results of the closure scheme are corroborated by numerical simulations.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Benchmarking inverse statistical approaches for protein structure and design with exactly solvable models

Inverse statistical approaches to determine protein structure and function from Multiple Sequence Alignments (MSA) are emerging as powerful tools in computational biology. However the underlying assumptions of the relationship between the inferred effective Potts Hamiltonian and real protein structure and energetics remain untested so far. Here we use lattice protein model (LP) to benchmark those inverse statistical approaches. We build MSA of highly stable sequences in target LP structures, and infer the effective pairwise Potts Hamiltonians from those MSA. We find that inferred Potts Hamiltonians reproduce many important aspects of 'true' LP structures and energetics. Careful analysis reveals that effective pairwise couplings in inferred Potts Hamiltonians depend not only on the energetics of the native structure but also on competing folds; in particular, the coupling values reflect both positive design (stabilization of native conformation) and negative design (destabilization of competing folds). In addition to providing detailed structural information, the inferred Potts models used as protein Hamiltonian for design of new sequences are able to generate with high probability completely new sequences with the desired folds, which is not possible using independent-site models. Those are remarkable results as the effective LP Hamiltonians used to generate MSA are not simple pairwise models due to the competition between the folds. Our findings elucidate the reasons for the success of inverse approaches to the modelling of proteins from sequence data, and their limitations.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Inference of principal components of noisy correlation matrices with prior information

The problem of infering the top component of a noisy sample covariance matrix with prior information about the distribution of its entries is considered, in the framework of the spiked covariance model. Using the replica method of statistical physics the computation of the overlap between the top components of the sample and population covariance matrices is formulated as an explicit optimization problem for any kind of entry-wise prior information. The approach is illustrated on the case of top components including large entries, and the corresponding phase diagram is shown. The calculation predicts that the maximal sampling noise level at which the recovery of the top population component remains possible is higher than its counterpart in the spiked covariance model with no prior information.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Estimating the principal components of correlation matrices from all their empirical eigenvectors

We consider the problem of estimating the principal components of a population correlation matrix from a limited number of measurement data. Using a combination of random matrix and information-theoretic tools, we show that all the eigenmodes of the sample correlation matrices are informative, and not only the top ones. We show how this information can be exploited when prior information about the principal component, such as whether it is localized or not, is available by mapping the estimation problem onto the search for the ground state of a spin-glass-like effective Hamiltonian encoding the prior. Results are illustrated numerically on the spiked covariance model.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Learning probabilities from random observables in high dimensions: the maximum entropy distribution and others

We consider the problem of learning a target probability distribution over a set of $N$ binary variables from the knowledge of the expectation values (with this target distribution) of $M$ observables, drawn uniformly at random. The space of all probability distributions compatible with these $M$ expectation values within some fixed accuracy, called version space, is studied. We introduce a biased measure over the version space, which gives a boost increasing exponentially with the entropy of the distributions and with an arbitrary inverse `temperature' $Γ$. The choice of $Γ$ allows us to interpolate smoothly between the unbiased measure over all distributions in the version space ($Γ=0$) and the pointwise measure concentrated at the maximum entropy distribution ($Γ\to \infty$). Using the replica method we compute the volume of the version space and other quantities of interest, such as the distance $R$ between the target distribution and the center-of-mass distribution over the version space, as functions of $α=(\log M)/N$ and $Γ$ for large $N$. Phase transitions at critical values of $α$ are found, corresponding to qualitative improvements in the learning of the target distribution and to the decrease of the distance $R$. However, for fixed $α$, the distance $R$ does not vary with $Γ$, which means that the maximum entropy distribution is not closer to the target distribution than any other distribution compatible with the observable values. Our results are confirmed by Monte Carlo sampling of the version space for small system sizes ($N\le 10$).

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

On the entropy of protein families

Proteins are essential components of living systems, capable of performing a huge variety of tasks at the molecular level, such as recognition, signalling, copy, transport, ... The protein sequences realizing a given function may largely vary across organisms, giving rise to a protein family. Here, we estimate the entropy of those families based on different approaches, including Hidden Markov Models used for protein databases and inferred statistical models reproducing the low-order (1-and 2-point) statistics of multi-sequence alignments. We also compute the entropic cost, that is, the loss in entropy resulting from a constraint acting on the protein, such as the fixation of one particular amino-acid on a specific site, and relate this notion to the escape probability of the HIV virus. The case of lattice proteins, for which the entropy can be computed exactly, allows us to provide another illustration of the concept of cost, due to the competition of different folds. The relevance of the entropy in relation to directed evolution experiments is stressed.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Collective motion of the activity (II)

The dynamics of a neural model for hippocampal place cells storing spatial maps is studied. In the absence of external input, depending on the number of cells and on the values of control parameters (number of environments stored, level of neural noise, average level of activity, connectivity of place cells), a 'clump' of spatially-localized activity can diffuse, or remains pinned due to crosstalk between the environments. In the single-environment case, the macroscopic coefficient of diffusion of the clump and its effective mobility are calculated analytically from first principles, and corroborated by numerical simulations. In the multi-environment case the heights and the widths of the pinning barriers are analytically characterized with the replica method; diffusion within one map is then in competition with transitions between different maps. Possible mechanisms enhancing mobility are proposed and tested.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Cross-talk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: phase diagram (I)

We study the stable phases of an attractor neural network model, with binary units, for hippocampal place cells encoding 1D or 2D spatial maps or environments. Using statistical mechanics tools we show that, below critical values for the noise in the neural response and for the number of environments, the network activity is spatially localized in one environment. We calculate the number of stored environments. For high noise and loads the network activity extends over space, either uniformly or with spatial heterogeneities due to the cross-talk between the maps, and memory of environments is lost. Analytical predictions are corroborated by numerical simulations.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Adaptive Cluster Expansion for Inferring Boltzmann Machines with Noisy Data

We introduce a procedure to infer the interactions among a set of binary variables, based on their sampled frequencies and pairwise correlations. The algorithm builds the clusters of variables contributing most to the entropy of the inferred Ising model, and rejects the small contributions due to the sampling noise. Our procedure successfully recovers benchmark Ising models even at criticality and in the low temperature phase, and is applied to neurobiological data.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Adaptive cluster expansion for the inverse Ising problem: convergence, algorithm and tests

We present a procedure to solve the inverse Ising problem, that is to find the interactions between a set of binary variables from the measure of their equilibrium correlations. The method consists in constructing and selecting specific clusters of variables, based on their contributions to the cross-entropy of the Ising model. Small contributions are discarded to avoid overfitting and to make the computation tractable. The properties of the cluster expansion and its performances on synthetic data are studied. To make the implementation easier we give the pseudo-code of the algorithm.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

On the trajectories and performance of Infotaxis, an information-based greedy search algorithm

We present a continuous-space version of Infotaxis, a search algorithm where a searcher greedily moves to maximize the gain in information about the position of the target to be found. Using a combination of analytical and numerical tools we study the nature of the trajectories in two and three dimensions. The probability that the search is successful and the running time of the search are estimated. A possible extension to non-greedy search is suggested.

0 citations0 reviewsNo code linkedOpen access
preprint2006arXiv

Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.

0 citations0 reviewsNo code linkedOpen access
preprint2006arXiv

Criticality and Universality in the Unit-Propagation Search Rule

The probability Psuccess(alpha, N) that stochastic greedy algorithms successfully solve the random SATisfiability problem is studied as a function of the ratio alpha of constraints per variable and the number N of variables. These algorithms assign variables according to the unit-propagation (UP) rule in presence of constraints involving a unique variable (1-clauses), to some heuristic (H) prescription otherwise. In the infinite N limit, Psuccess vanishes at some critical ratio alpha\_H which depends on the heuristic H. We show that the critical behaviour is determined by the UP rule only. In the case where only constraints with 2 and 3 variables are present, we give the phase diagram and identify two universality classes: the power law class, where Psuccess[alpha\_H (1+epsilon N^{-1/3}), N] ~ A(epsilon)/N^gamma; the stretched exponential class, where Psuccess[alpha\_H (1+epsilon N^{-1/3}), N] ~ exp[-N^{1/6} Phi(epsilon)]. Which class is selected depends on the characteristic parameters of input data. The critical exponent gamma is universal and calculated; the scaling functions A and Phi weakly depend on the heuristic H and are obtained from the solutions of reaction-diffusion equations for 1-clauses. Computation of some non-universal corrections allows us to match numerical results with good precision. The critical behaviour for constraints with >3 variables is given. Our results are interpreted in terms of dynamical graph percolation and we argue that they should apply to more general situations where UP is used.

0 citations0 reviewsNo code linkedOpen access
preprint2005arXiv

Field theoretic approach to metastability in the contact process

A quantum field theoretic formulation of the dynamics of the Contact Process on a regular graph of degree z is introduced. A perturbative calculation in powers of 1/z of the effective potential for the density of particles phi(t) and an instantonic field psi(t) emerging from the quantum formalism is performed. Corrections to the mean-field distribution of densities of particles in the out-of-equilibrium stationary state are derived in powers of 1/z. Results for typical (e.g. average density) and rare fluctuation (e.g. lifetime of the metastable state) properties are in very good agreement with numerical simulations carried out on D-dimensional hypercubic (z=2D) and Cayley lattices.

0 citations0 reviewsNo code linkedOpen access