arXiv subjects

Manon Michel

Publications and source records attributed to Manon Michel.

At least 19 recordsLinked to original sources

arXiv2026

Event-Chain Monte Carlo for Yang-Mills SU(N) lattice field theory I : Design and proof of concept

We develop two implementations of the Event-Chain Monte Carlo (ECMC) algorithm for Yang-Mills $\mathrm{SU}(N)$ lattice gauge theories with the Wilson action. These algorithms consist in a succession of local ballistic updates intersped with stochastic events, resulting in an irreversible and rejection-free Markov process. The resulting dynamics satisfy global balance, ensuring the correct equilibrium distribution. The algorithms are formulated for general $\mathrm{SU}(N)$ Yang-Mills theories with Wilson action and implemented for the case $N=3$. Numerical tests on four-dimensional lattices show that standard gauge observables, such as the mean plaquette, agree with results obtained using conventional Monte Carlo algorithms. These results provide a first validation of ECMC as a viable sampling scheme for Yang-Mills lattice gauge theories.

hep-lat↗

arXiv2026

Quantifying Uncertainty In Wide Two-Layer Neural Networks: On The Law Of The Limiting Fluctuation Process

Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.

cs.NE↗

arXiv2025

Activity-driven clustering and many-body steady state of jamming run-and-tumble particles

We exactly resolve the three-particle steady state of run-and-tumble particles with jamming interactions, providing the first microscopic description beyond two bodies. The invariant measure, derived via a piecewise-deterministic Markov process description and symmetry principles, reveals persistent, separated, and diffusive regimes ruled by the activity parameter. A geometric cascade of scales in the activity parameter organizes the structural weights, showing the separated phase dominates at finite activity, while non-uniformity plays only a minor role. Extending these results to larger systems, we show that the $N$-body steady state inherits the same organization: the number of clusters becomes sharply defined by the activity value, with crossover boundaries whose slopes diverge with $N$. We also show how the activity plays a role similar to a fugacity conjugate to cluster number, yielding a grand-canonical-like structure emerging directly from the microscopic dynamics. This framework lays the groundwork for a systematic microscopic theory of active many-body steady states.

cond-mat.stat-mech↗

arXiv2025

Bosonized one-dimensional quantum systems through enhanced event-chain Monte Carlo

We design an enhanced Event-Chain Monte Carlo algorithm to study 1D quantum dissipative systems, using their bosonized representation. Expressing the bosonized Hamiltonian as a path integral over a scalar field enables the application of Monte Carlo algorithms developed for classical systems. Specifically, we focus on a dissipative XXZ spin chain, exhibiting critical slowing down, minima degeneracy and long-range interactions. Addressing all three bottlenecks, we design an algorithm that combines local persistent Event-Chain Monte Carlo moves with global cluster moves, in a O(1)-complexity implementation. Through systematic performance analysis, we show that such an algorithm outperforms traditional Metropolis algorithms by more than a magnitude factor and is competitive with current state-of-the-art Quantum Monte Carlo algorithms. We then use this approach to determine the dissipative spin chain's phase diagram, thereby reinforcing prior analytical predictions.

cond-mat.str-el↗

arXiv2025

Convergence of non-reversible Markov processes via lifting and flow Poincar{\'e} inequality

We propose a general approach for quantitative convergence analysis of non-reversible Markov processes, based on the concept of second-order lifts and a variational approach to hypocoercivity. To this end, we introduce the flow Poincar{\'e} inequality, a space-time Poincar{\'e} inequality along trajectories of the semigroup, and a general divergence lemma based only on the Dirichlet form of an underlying reversible diffusion. We demonstrate the versatility of our approach by applying it to a pair of run-and-tumble particles with jamming, a model from non-equilibrium statistical mechanics, and several piecewise deterministic Markov processes used in sampling applications, in particular including general stochastic jump kernels.

math.AP↗

arXiv2024

Long-time analysis of a pair of on-lattice and continuous run-and-tumble particles with jamming interactions

Run-and-Tumble Particles (RTPs) are a key model of active matter. They are characterized by alternating phases of linear travel and random direction reshuffling. By this dynamic behavior, they break time reversibility and energy conservation at the microscopic level. It leads to complex out-of-equilibrium phenomena such as collective motion, pattern formation, and motility-induced phase separation (MIPS). In this work, we study two fundamental dynamical models of a pair of RTPs with jamming interactions and provide a rigorous link between their discrete- and continuous-space descriptions. We demonstrate that as the lattice spacing vanishes, the discrete models converge to a continuous RTP model on the torus, described by a Piecewise Deterministic Markov Process (PDMP). This establishes that the invariant measures of the discrete models converge to that of the continuous model, which reveals finite mass at jamming configurations and exponential decay away from them. This indicates effective attraction, which is consistent with MIPS. Furthermore, we quantitatively explore the convergence towards the invariant measure. Such convergence study is critical for understanding and characterizing how MIPS emerges over time. Because RTP systems are non-reversible, usual methods may fail or are limited to qualitative results. Instead, we adopt a coupling approach to obtain more accurate, non-asymptotic bounds on mixing times. The findings thus provide deeper theoretical insights into the mixing times of these RTP systems, revealing the presence of both persistent and diffusive regimes.

math.PR↗

arXiv2024

Central Limit Theorem for Bayesian Neural Network trained with Variational Inference

In this paper, we rigorously derive Central Limit Theorems (CLT) for Bayesian two-layerneural networks in the infinite-width limit and trained by variational inference on a regression task. The different networks are trained via different maximization schemes of the regularized evidence lower bound: (i) the idealized case with exact estimation of a multiple Gaussian integral from the reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling, commonly known as Bayes-by-Backprop, and (iii) a computationally cheaper algorithm named Minimal VI. The latter was recently introduced by leveraging the information obtained at the level of the mean-field limit. Laws of large numbers are already rigorously proven for the three schemes that admits the same asymptotic limit. By deriving CLT, this work shows that the idealized and Bayes-by-Backprop schemes have similar fluctuation behavior, that is different from the Minimal VI one. Numerical experiments then illustrate that the Minimal VI scheme is still more efficient, in spite of bigger variances, thanks to its important gain in computational complexity.

stat.ML↗

arXiv2023

Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference

We provide a rigorous analysis of training by variational inference (VI) of Bayesian neural networks in the two-layer and infinite-width case. We consider a regression problem with a regularized evidence lower bound (ELBO) which is decomposed into the expected log-likelihood of the data and the Kullback-Leibler (KL) divergence between the a priori distribution and the variational posterior. With an appropriate weighting of the KL, we prove a law of large numbers for three different training schemes: (i) the idealized case with exact estimation of a multiple Gaussian integral from the reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling, commonly known as Bayes by Backprop, and (iii) a new and computationally cheaper algorithm which we introduce as Minimal VI. An important result is that all methods converge to the same mean-field limit. Finally, we illustrate our results numerically and discuss the need for the derivation of a central limit theorem.

stat.ML↗

arXiv2023

Necessary and sufficient symmetries in Event-Chain Monte Carlo with generalized flows and Application to hard dimers

Event-Chain Monte Carlo (ECMC) methods generate continuous-time and non-reversible Markov processes which often display significant accelerations compared to reversible counterparts. However their generalization to any system may appear less straightforward. In this work, our aim is to distinctly define the essential symmetries that such ECMC algorithms must adhere to, differentiating between necessary and sufficient conditions. This exploration intends to delineate the balance between requirements that could be overly limiting in broad applications and those that are fundamentally essential. To do so, we build on the recent analytical description of such methods as generating Piecewise Deterministic Markov Processes (PDMP). Thus, starting with translational flows, we establish the necessary rotational invariance of the probability flows, along with determining the minimum event rate. This rate identifies with the corresponding infinitesimal Metropolis rejection rate. Obeying such conditions ensures the correct invariance for any ECMC scheme. Subsequently, we extend these findings to encompass schemes involving deterministic flows that are more general than mere translational ones. Specifically, we define two classes of interest of general flows: the ideal and uniform-ideal ones. They respectively suppresses or reduces the event rates. From there, we implement a comprehensive non-reversible sampling of a systems of hard dimers by introducing rotational flows, which are uniform-ideal. This implementation results in a speed-up of up to $\sim 3$ compared to the state-of-the-art ECMC/Metropolis hybrid scheme.

cond-mat.stat-mech↗

arXiv2023

Jamming pair of general run-and-tumble particles: Exact results, symmetries and steady-state universality classes

While run-and-tumble particles are a foundational model for self-propelled particles as bacteria or Janus particles, the analytical derivation of their steady state from the microscopic details is still an open problem. By directly modeling the system at the continuous-space and -time level thanks to piecewise deterministic Markov processes (PDMP), we derive the conservation conditions which sets the invariant distribution and, more importantly, explicitly construct the two universality classes for the steady state, the detailed-jamming and the global-jamming classes. They respectively identify with the preservation or not in a detailed manner of a symmetry at the level of the dynamical internal states between probability flows entering and exiting jamming configurations. We call such symmetry active global balance, as it is the true nonequilibrium counterpart of the equilibrium global balance. Thanks to a spectral analysis of the tumble kernel, we give explicit expressions for the invariant measure in the general case. We show that the non-equilibrium features exhibited by the steady state include positive mass for the jammed configurations and, for the global-jamming class, exponential decay and growth terms, potentially modulated by polynomial terms. Interestingly, we find that the invariant measure follows, away from jamming configurations, a catenary-like constraint, which results from the interplay between probability conservation and the dynamical skewness introduced by the jamming interactions, seen now as a boundary constraint. This work shows the powerful analytical approach PDMP provide for the study of the stationary behaviors of RTP systems and motivates their future applications to larger systems, with the goal to derive microscopic conditions for motility-induced phase transitions.

cond-mat.stat-mech↗

arXiv2023

Law of large numbers and central limit theorem for wide two-layer neural networks: the mini-batch and noisy case

In this work, we consider a wide two-layer neural network and study the behavior of its empirical weights under a dynamics set by a stochastic gradient descent along the quadratic loss with mini-batches and noise. Our goal is to prove a trajectorial law of large number as well as a central limit theorem for their evolution. When the noise is scaling as 1/N $β$ and 1/2 < $β$ $\le$ $\infty$, we rigorously derive and generalize the LLN obtained for example in [CRBVE20, MMM19, SS20b]. When 3/4 < $β$ $\le$ $\infty$, we also generalize the CLT (see also [SS20a]) and further exhibit the effect of mini-batching on the asymptotic variance which leads the fluctuations. The case $β$ = 3/4 is trickier and we give an example showing the divergence with time of the variance thus establishing the instability of the predictions of the neural network in this case. It is illustrated by simple numerical examples.

math.PR↗

arXiv2023

Fixed-kinetic Neural Hamiltonian Flows for enhanced interpretability and reduced complexity

Normalizing Flows (NF) are Generative models which transform a simple prior distribution into the desired target. They however require the design of an invertible mapping whose Jacobian determinant has to be computable. Recently introduced, Neural Hamiltonian Flows (NHF) are Hamiltonian dynamics-based flows, which are continuous, volume-preserving and invertible and thus make for natural candidates for robust NF architectures. In particular, their similarity to classical Mechanics could lead to easier interpretability of the learned mapping. In this paper, we show that the current NHF architecture may still pose a challenge to interpretability. Inspired by Physics, we introduce a fixed-kinetic energy version of the model. This approach improves interpretability and robustness while requiring fewer parameters than the original model. We illustrate that on a 2D Gaussian mixture and on the MNIST and Fashion-MNIST datasets. Finally, we show how to adapt NHF to the context of Bayesian inference and illustrate the method on an example from cosmology.

cs.LG↗

arXiv2022

PDMP characterisation of event-chain Monte Carlo algorithms for particle systems

Monte Carlo simulations of systems of particles such as hard spheres or soft spheres with singular kernels can display around a phase transition prohibitively long convergence times when using traditional Hasting-Metropolis reversible schemes. Efficient algorithms known as event-chain Monte Carlo were then developed to reach necessary accelerations. They are based on non-reversible continuous-time Markov processes. Proving invariance and ergodicity for such schemes cannot be done as for discrete-time schemes and a theoretical framework to do so was lacking, impeding the generalisation of ECMC algorithms to more sophisticated systems or processes. In this work, we characterize the Markov processes generated in ECMC as piecewise deterministic Markov processes. It first allows us to propose more general schemes, for instance regarding the direction refreshment. We then prove the invariance of the correct stationary distribution. Finally, we show the ergodicity of the processes in soft- and hard-sphere systems, with a density condition for the latter.

cond-mat.stat-mech↗

arXiv2020

Loop-Cluster Coupling and Algorithm for Classical Statistical Models

Potts spin systems play a fundamental role in statistical mechanics and quantum field theory, and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the $q$-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with $q$-flow variables, and formulate a LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the $q$-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model, but also brings new insights into rich geometric structures of the FK clusters.

cond-mat.stat-mech↗

arXiv2020

Forward Event-Chain Monte Carlo: Fast sampling by randomness control in irreversible Markov chains

Irreversible and rejection-free Monte Carlo methods, recently developed in Physics under the name Event-Chain and known in Statistics as Piecewise Deterministic Monte Carlo (PDMC), have proven to produce clear acceleration over standard Monte Carlo methods, thanks to the reduction of their random-walk behavior. However, while applying such schemes to standard statistical models, one generally needs to introduce an additional randomization for sake of correctness. We propose here a new class of Event-Chain Monte Carlo methods that reduces this extra-randomization to a bare minimum. We compare the efficiency of this new methodology to standard PDMC and Monte Carlo methods. Accelerations up to several magnitudes and reduced dimensional scalings are exhibited.

stat.CO↗

arXiv2019

Clock Monte Carlo methods

We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of ${\rm O}(N^κ)$ ($0 \! \leq \! κ\! \leq \! 1$) can be achieved compared to the Metropolis method, where $N$ is the system size and the $κ$ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O$(n)$ spin models in a wide parameter regime: for $n \! = \! 1,2,3$, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from $d \! = \! 1, 2, 3$ to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.

cond-mat.stat-mech↗

arXiv2017

Event-chain Monte Carlo algorithms for three- and many-particle interactions

We generalize the rejection-free event-chain Monte Carlo algorithm from many particle systems with pairwise interactions to systems with arbitrary three- or many-particle interactions. We introduce generalized lifting probabilities between particles and obtain a general set of equations for lifting probabilities, the solution of which guarantees maximal global balance. We validate the resulting three-particle event-chain Monte Carlo algorithms on three different systems by comparison with conventional local Monte Carlo simulations: (i) a test system of three particles with a three-particle interaction that depends on the enclosed triangle area; (ii) a hard-needle system in two dimensions, where needle interactions constitute three-particle interactions of the needle end points; (iii) a semiflexible polymer chain with a bending energy, which constitutes a three-particle interaction of neighboring chain beads. The examples demonstrate that the generalization to many-particle interactions broadens the applicability of event-chain algorithms considerably.

cond-mat.stat-mech↗

arXiv2015

Event-chain algorithm for the Heisenberg model: Evidence for $z \simeq 1$ dynamic scaling

We apply the event-chain Monte Carlo algorithm to the three-dimensional ferromagnetic Heisenberg model. The algorithm is rejection-free and also realizes an irreversible Markov chain that satisfies global balance. The autocorrelation functions of the magnetic susceptibility and the energy indicate a dynamical critical exponent $z \approx 1$ at the critical temperature, while that of the magnetization does not measure the performance of the algorithm. This seems to be the first report that the event-chain Monte Carlo algorithm substantially reduces the dynamical critical exponent from the conventional value of $z\simeq 2$.

cond-mat.stat-mech↗