Restricted Boltzmann Machines (RBMs) are powerful tools for modeling complex systems and extracting insights from data, but their training is hindered by the slow mixing of Markov Chain Monte Carlo (MCMC) processes, especially with highly structured datasets. In this study, we build on recent theoretical advances in RBM training and focus on the stepwise encoding of data patterns into singular vectors of the coupling matrix, significantly reducing the cost of generating new samples and evaluating the quality of the model, as well as the training cost in highly clustered datasets. The learning process is analogous to the thermodynamic continuous phase transitions observed in ferromagnetic models, where new modes in the probability measure emerge in a continuous manner. We leverage the continuous transitions in the training process to define a smooth annealing trajectory that enables reliable and computationally efficient log-likelihood estimates. This approach enables online assessment during training and introduces a novel sampling strategy called Parallel Trajectory Tempering (PTT) that outperforms previously optimized MCMC methods. To mitigate the critical slowdown effect in the early stages of training, we propose a pre-training phase. In this phase, the principal components are encoded into a low-rank RBM through a convex optimization process, facilitating efficient static Monte Carlo sampling and accurate computation of the partition function. Our results demonstrate that this pre-training strategy allows RBMs to efficiently handle highly structured datasets where conventional methods fail. Additionally, our log-likelihood estimation outperforms computationally intensive approaches in controlled scenarios, while the PTT algorithm significantly accelerates MCMC processes compared to conventional methods.

In the recent years, Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems, in particular for data assimilation purpose. This method is still in its infancy, with many shortcomings and failures that remain not properly understood. In this paper we propose a natural gradient approach to PINNs which contributes to speed-up and improve the accuracy of the training. Based on an in depth analysis of the differential geometric structures of the problem, we come up with two distinct contributions: (i) a new natural gradient algorithm that scales as $\min(P^2S, S^2P)$, where $P$ is the number of parameters, and $S$ the batch size; (ii) a mathematically principled reformulation of the PINNs problem that allows the extension of natural gradient to it, with proved connections to Green's function theory.

We investigate the impact of limited data on training pairwise energy-based models for inverse problems aimed at identifying interaction networks. Utilizing the Gaussian model as testbed, we dissect training trajectories across the eigenbasis of the coupling matrix, exploiting the independent evolution of eigenmodes and revealing that the learning timescales are tied to the spectral decomposition of the empirical covariance matrix. We see that optimal points for early stopping arise from the interplay between these timescales and the initial conditions of training. Moreover, we show that finite data corrections can be accurately modeled through asymptotic random matrix theory calculations and provide the counterpart of generalized cross-validation in the energy based model context. Our analytical framework extends to binary-variable maximum-entropy pairwise models with minimal variations. These findings offer strategies to control overfitting in discrete-variable models through empirical shrinkage corrections, improving the management of overfitting in energy-based generative models. Finally, we propose a generalization to arbitrary energy-based models by deriving the neural tangent kernel dynamics of the score function under the score-matching algorithm.

Applications of generative models for genomic data have gained significant momentum in the past few years, with scopes ranging from data characterization to generation of genomic segments and functional sequences. In our previous study, we demonstrated that generative adversarial networks (GANs) and restricted Boltzmann machines (RBMs) can be used to create novel high-quality artificial genomes (AGs) which can preserve the complex characteristics of real genomes such as population structure, linkage disequilibrium and selection signals. However, a major drawback of these models is scalability, since the large feature space of genome-wide data increases computational complexity vastly. To address this issue, we implemented a novel convolutional Wasserstein GAN (WGAN) model along with a novel conditional RBM (CRBM) framework for generating AGs with high SNP number. These networks implicitly learn the varying landscape of haplotypic structure in order to capture complex correlation patterns along the genome and generate a wide diversity of plausible haplotypes. We performed comparative analyses to assess both the quality of these generated haplotypes and the amount of possible privacy leakage from the training data. As the importance of genetic privacy becomes more prevalent, the need for effective privacy protection measures for genomic data increases. We used generative neural networks to create large artificial genome segments which possess many characteristics of real genomes without substantial privacy leakage from the training dataset. In the near future, with further improvements in haplotype quality and privacy preservation, large-scale artificial genome databases can be assembled to provide easily accessible surrogates of real databases, allowing researchers to conduct studies with diverse genomic data within a safe ethical framework in terms of donor privacy.

The probabilistic forecasting method described in this study is devised to leverage spatial and temporal dependency of urban traffic networks, in order to provide predictions accurate at short term and meaningful for a horizon of up to several hours. By design, it can deal with missing data, both for training and running the model. It is able to forecast the state of the entire network in one pass, with an execution time that scales linearly with the size of the network. The method consists in learning a sparse Gaussian copula of traffic variables, compatible with the Gaussian belief propagation algorithm. The model is trained automatically from an historical dataset through an iterative proportional scaling procedure, that is well suited to compatibility constraints induced by Gaussian belief propagation. Results of tests performed on two urban datasets show a very good ability to predict flow variables and reasonably good performances on speed variables. Some understanding of the observed performances is given by a careful analysis of the model, making it to some degree possible to disentangle modeling bias from the intrinsic noise of the traffic phenomena and its measurement process.

Training Restricted Boltzmann Machines (RBMs) has been challenging for a long time due to the difficulty of computing precisely the log-likelihood gradient. Over the past decades, many works have proposed more or less successful recipes but without studying systematically the crucial quantity of the problem: the mixing time i.e. the number of MCMC iterations needed to sample completely new configurations from a model. In this work, we show that this mixing time plays a crucial role in the behavior and stability of the trained model, and that RBMs operate in two well-defined distinct regimes, namely equilibrium and out-of-equilibrium, depending on the interplay between this mixing time of the model and the number of MCMC steps, $k$, used to approximate the gradient. We further show empirically that this mixing time increases along the learning, which often implies a transition from one regime to another as soon as $k$ becomes smaller than this time.In particular, we show that using the popular $k$ (persistent) contrastive divergence approaches, with $k$ small, the dynamics of the fitted model are extremely slow and often dominated by strong out-of-equilibrium effects. On the contrary, RBMs trained in equilibrium display much faster dynamics, and a smooth convergence to dataset-like configurations during the sampling.Finally, we discuss how to exploit in practice both regimes depending on the task one aims to fulfill: (i) short $k$s can be used to generate convincing samples in short learning times, (ii) large $k$ (or increasingly large) must be used to learn the correct equilibrium distribution of the RBM. Finally, the existence of these two operational regimes seems to be a general property of energy based models trained via likelihood maximization.

Data stream clustering provides insights into the underlying patterns of data flows. This paper focuses on selecting the best representatives from clusters of streaming data. There are two main challenges: how to cluster with the best representatives and how to handle the evolving patterns that are important characteristics of streaming data with dynamic distributions. We employ the Affinity Propagation (AP) algorithm presented in 2007 by Frey and Dueck for the first challenge, as it offers good guarantees of clustering optimality for selecting exemplars. The second challenging problem is solved by change detection. The presented StrAP algorithm combines AP with a statistical change point detection test; the clustering model is rebuilt whenever the test detects a change in the underlying data distribution. Besides the validation on two benchmark data sets, the presented algorithm is validated on a real-world application, monitoring the data flow of jobs submitted to the EGEE grid.

The Affinity Propagation (AP) clustering algorithm proposed by Frey and Dueck (2007) provides an understandable, nearly optimal summary of a dataset, albeit with quadratic computational complexity. This paper, motivated by Autonomic Computing, extends AP to the data streaming framework. Firstly a hierarchical strategy is used to reduce the complexity to O(N1+ε); the distortion loss incurred is analyzed in relation with the dimension of the data items. Secondly, a coupling with a change detection test is used to cope with non-stationary data distribution, and rebuild the model as needed. The presented approach StrAP is applied to the stream of jobs submitted to the EGEE Grid, providing an understandable description of the job flow and enabling the system administrator to spot online some sources of failures.