Omar Chehab

Omar Chehab, Alexandre Gramfort, Aapo Hyvarinen

Learning a parametric model of a data distribution is a well-known statistical problem that has seen renewed interest as it is brought to scale in deep learning. Framing the problem as a self-supervised task, where data samples are discriminated from noise samples, is at the core of state-of-the-art methods, beginning with Noise-Contrastive Estimation (NCE). Yet, such contrastive learning requires a good noise distribution, which is hard to specify; domain-specific heuristics are therefore widely used. While a comprehensive theory is missing, it is widely assumed that the optimal noise should in practice be made equal to the data, both in distribution and proportion. This setting underlies Generative Adversarial Networks (GANs) in particular. Here, we empirically and theoretically challenge this assumption on the optimal noise. We show that deviating from this assumption can actually lead to better statistical estimators, in terms of asymptotic variance. In particular, the optimal noise distribution is different from the data's and even from a different family.

Omar Chehab, Alexandre Gramfort, Aapo Hyvarinen

Self-supervised learning is an increasingly popular approach to unsupervised learning, achieving state-of-the-art results. A prevalent approach consists in contrasting data points and noise points within a classification task: this requires a good noise distribution which is notoriously hard to specify. While a comprehensive theory is missing, it is widely assumed that the optimal noise distribution should in practice be made equal to the data distribution, as in Generative Adversarial Networks (GANs). We here empirically and theoretically challenge this assumption. We turn to Noise-Contrastive Estimation (NCE) which grounds this self-supervised task as an estimation problem of an energy-based model of the data. This ties the optimality of the noise distribution to the sample efficiency of the estimator, which is rigorously defined as its asymptotic variance, or mean-squared error. In the special case where the normalization constant only is unknown, we show that NCE recovers a family of Importance Sampling estimators for which the optimal noise is indeed equal to the data distribution. However, in the general case where the energy is also unknown, we prove that the optimal noise density is the data density multiplied by a correction term based on the Fisher score. In particular, the optimal noise distribution is different from the data distribution, and is even from a different family. Nevertheless, we soberly conclude that the optimal noise may be hard to sample from, and the gain in efficiency can be modest compared to choosing the noise distribution equal to the data's.

Omar Chehab, Aapo Hyvarinen, Andrej Risteski

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

Hanlin Yu, RuiKang OuYang, Partha Kaushik et al.

Learning an energy-based model from data samples is a central problem in machine learning. Many recent and popular methods, such as denoising score matching for training energy-based diffusion models, use stochastic interpolants to corrupt data samples at different noise levels indexed by a time variable. This defines a joint density over both the data space and time, and most methods learn its energy through either spatial or temporal differences. We identify distinct failure modes for both of these approaches. To solve them, we propose Spatiotemporal Noise-Contrastive Estimation (stNCE), a framework for learning the energy through joint spatiotemporal differences. stNCE unifies many existing methods and leads to new training objectives. Experiments on images and molecules demonstrate performance competitive with state-of-the-art density estimation methods.

Siqiao Huang, Partha Kaushik, Michael Chen et al.

World models have become a central paradigm for learning predictive simulators that support generation, planning, and decision-making. Yet, despite rapid progress in industry-scale interactive video generation, the broader research community still lacks compact, reproducible, and easily extensible implementations for studying the design choices underlying modern world models. We introduce Nano World Models, a minimalist codebase for future video prediction centered around diffusion forcing. Nano World Models provides a unified interface for generative objectives, model scales, action-conditioning mechanisms, latent observation spaces, datasets, evaluation protocols, and long-horizon rollout procedures. This design enables controlled studies of world-modeling components that are often entangled across separate implementations. Through experiments across simple control environments, game simulation, and real-robot data, we examine how prediction parameterization, architecture scale, action injection, sampling budget, and domain complexity affect video prediction quality and autoregressive rollout behavior. By releasing code, configurations, evaluation scripts, and pretrained checkpoints, Nano World Models aims to provide a compact yet extensible experimental substrate for open, reproducible, and scientific world-model research.

Omar Chehab, Anna Korba, Austin Stromme et al.

Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that geometric tempering may not help, and can even be harmful for convergence.

Hanlin Yu, Arto Klami, Aapo Hyvärinen et al.

Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

Adrien Vacher, Omar Chehab, Anna Korba

Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions -- including all Gaussian mixtures -- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.

Ambroise Heurtebise, Omar Chehab, Pierre Ablin et al.

Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.

Ambroise Heurtebise, Omar Chehab, Pierre Ablin et al.

Machine learning techniques in multi-view settings face significant challenges, particularly when integrating heterogeneous data, aligning feature spaces, and managing view-specific biases. These issues are prominent in neuroscience, where data from multiple subjects exposed to the same stimuli are analyzed to uncover brain activity dynamics. In magnetoencephalography (MEG), where signals are captured at the scalp level, estimating the brain's underlying sources is crucial, especially in group studies where sources are assumed to be similar for all subjects. Common methods, such as Multi-View Independent Component Analysis (MVICA), assume identical sources across subjects, but this assumption is often too restrictive due to individual variability and age-related changes. Multi-View Independent Component Analysis with Delays (MVICAD) addresses this by allowing sources to differ up to a temporal delay. However, temporal dilation effects, particularly in auditory stimuli, are common in brain dynamics, making the estimation of time delays alone insufficient. To address this, we propose Multi-View Independent Component Analysis with Delays and Dilations (MVICAD2), which allows sources to differ across subjects in both temporal delays and dilations. We present a model with identifiable sources, derive an approximation of its likelihood in closed form, and use regularization and optimization techniques to enhance performance. Through simulations, we demonstrate that MVICAD2 outperforms existing multi-view ICA methods. We further validate its effectiveness using the Cam-CAN dataset, and showing how delays and dilations are related to aging.

Omar Chehab, Anna Korba

Diffusion models are state-of-the-art methods in generative modeling when samples from a target probability distribution are available, and can be efficiently sampled, using score matching to estimate score vectors guiding a Langevin process. However, in the setting where samples from the target are not available, e.g. when this target's density is known up to a normalization constant, the score estimation task is challenging. Previous approaches rely on Monte Carlo estimators that are either computationally heavy to implement or sample-inefficient. In this work, we propose a computationally attractive alternative, relying on the so-called dilation path, that yields score vectors that are available in closed-form. This path interpolates between a Dirac and the target distribution using a convolution. We propose a simple implementation of Langevin dynamics guided by the dilation path, using adaptive step-sizes. We illustrate the results of our sampling method on a range of tasks, and shows it performs better than classical alternatives.

Omar Chehab, Alexandre Defossez, Jean-Christophe Loiseau et al.

Understanding how the brain responds to sensory inputs is challenging: brain recordings are partial, noisy, and high dimensional; they vary across sessions and subjects and they capture highly nonlinear dynamics. These challenges have led the community to develop a variety of preprocessing and analytical (almost exclusively linear) methods, each designed to tackle one of these issues. Instead, we propose to address these challenges through a specific end-to-end deep learning architecture, trained to predict the brain responses of multiple subjects at once. We successfully test this approach on a large cohort of magnetoencephalography (MEG) recordings acquired during a one-hour reading task. Our Deep Recurrent Encoding (DRE) architecture reliably predicts MEG responses to words with a three-fold improvement over classic linear methods. To overcome the notorious issue of interpretability of deep learning, we describe a simple variable importance analysis. When applied to DRE, this method recovers the expected evoked responses to word length and word frequency. The quantitative improvement of the present deep learning approach paves the way to better understand the nonlinear dynamics of brain activity from large datasets.

Hubert Banville, Omar Chehab, Aapo Hyvärinen et al.

Objective. Supervised learning paradigms are often limited by the amount of labeled data that is available. This phenomenon is particularly problematic in clinically-relevant data, such as electroencephalography (EEG), where labeling can be costly in terms of specialized expertise and human processing time. Consequently, deep learning architectures designed to learn on EEG data have yielded relatively shallow models and performances at best similar to those of traditional feature-based approaches. However, in most situations, unlabeled data is available in abundance. By extracting information from this unlabeled data, it might be possible to reach competitive performance with deep neural networks despite limited access to labels. Approach. We investigated self-supervised learning (SSL), a promising technique for discovering structure in unlabeled data, to learn representations of EEG signals. Specifically, we explored two tasks based on temporal context prediction as well as contrastive predictive coding on two clinically-relevant problems: EEG-based sleep staging and pathology detection. We conducted experiments on two large public datasets with thousands of recordings and performed baseline comparisons with purely supervised and hand-engineered approaches. Main results. Linear classifiers trained on SSL-learned features consistently outperformed purely supervised deep neural networks in low-labeled data regimes while reaching competitive performance when all labels were available. Additionally, the embeddings learned with each method revealed clear latent structures related to physiological and clinical phenomena, such as age effects. Significance. We demonstrate the benefit of self-supervised learning approaches on EEG data. Our results suggest that SSL may pave the way to a wider use of deep learning models on EEG data.