Time series forecasting presents unique challenges that limit the effectiveness of traditional machine learning algorithms. To address these limitations, various approaches have incorporated linear constraints into learning algorithms, such as generalized additive models and hierarchical forecasting. In this paper, we propose a unified framework for integrating and combining linear constraints in time series forecasting. Within this framework, we show that the exact minimizer of the constrained empirical risk can be computed efficiently using linear algebra alone. This approach allows for highly scalable implementations optimized for GPUs. We validate the proposed methodology through extensive benchmarking on real-world tasks, including electricity demand forecasting and tourism forecasting, achieving state-of-the-art performance.

Denoising score matching plays a pivotal role in the performance of diffusion-based generative models. However, the empirical optimal score--the exact solution to the denoising score matching--leads to memorization, where generated samples replicate the training data. Yet, in practice, only a moderate degree of memorization is observed, even without explicit regularization. In this paper, we investigate this phenomenon by uncovering an implicit regularization mechanism driven by large learning rates. Specifically, we show that in the small-noise regime, the empirical optimal score exhibits high irregularity. We then prove that, when trained by stochastic gradient descent with a large enough learning rate, neural networks cannot stably converge to a local minimum with arbitrarily small excess risk. Consequently, the learned score cannot be arbitrarily close to the empirical optimal score, thereby mitigating memorization. To make the analysis tractable, we consider one-dimensional data and two-layer neural networks. Experiments validate the crucial role of the learning rate in preventing memorization, even beyond the one-dimensional setting.

Conformal Prediction (CP) is a principled framework for quantifying uncertainty in blackbox learning models, by constructing prediction sets with finite-sample coverage guarantees. Traditional approaches rely on scalar nonconformity scores, which fail to fully exploit the geometric structure of multivariate outputs, such as in multi-output regression or multiclass classification. Recent methods addressing this limitation impose predefined convex shapes for the prediction sets, potentially misaligning with the intrinsic data geometry. We introduce a novel CP procedure handling multivariate score functions through the lens of optimal transport. Specifically, we leverage Monge-Kantorovich vector ranks and quantiles to construct prediction region with flexible, potentially non-convex shapes, better suited to the complex uncertainty patterns encountered in multivariate learning tasks. We prove that our approach ensures finite-sample, distribution-free coverage properties, similar to typical CP methods. We then adapt our method for multi-output regression and multiclass classification, and also propose simple adjustments to generate adaptive prediction regions with asymptotic conditional coverage guarantees. Finally, we evaluate our method on practical regression and classification problems, illustrating its advantages in terms of (conditional) coverage and efficiency.

Attention-based models, such as Transformer, excel across various tasks but lack a comprehensive theoretical understanding, especially regarding token-wise sparsity and internal linear representations. To address this gap, we introduce the single-location regression task, where only one token in a sequence determines the output, and its position is a latent random variable, retrievable via a linear projection of the input. To solve this task, we propose a dedicated predictor, which turns out to be a simplified version of a non-linear self-attention layer. We study its theoretical properties, by showing its asymptotic Bayes optimality and analyzing its training dynamics. In particular, despite the non-convex nature of the problem, the predictor effectively learns the underlying structure. This work highlights the capacity of attention mechanisms to handle sparse token information and internal linear structures.

Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data-driven term and a partial differential equation (PDE) regularization. Building on the formulation of the problem as a kernel regression task, we use Fourier methods to approximate the associated kernel, and propose a tractable estimator that minimizes the physics-informed risk function. We refer to this approach as physics-informed kernel learning (PIKL). This framework provides theoretical guarantees, enabling the quantification of the physical prior's impact on convergence speed. We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time. Additionally, we identify cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions.

Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a partial differential equation that quantifies the physical inconsistency. We prove that for linear differential priors, the problem can be formulated as a kernel regression task. Taking advantage of kernel theory, we derive convergence rates for the minimizer of the regularized risk and show that it converges at least at the Sobolev minimax rate. However, faster rates can be achieved, depending on the physical error. This principle is illustrated with a one-dimensional example, supporting the claim that regularizing the empirical risk with physical information can be beneficial to the statistical performance of estimators.

Recent literature has focused extensively on assessing the error between the target and estimated distributions, gauging the generative quality through the Kullback-Leibler (KL) divergence and Wasserstein distances. All existing results have been obtained so far for time-homogeneous speed of the noise schedule. Under mild assumptions on the data distribution, we establish an upper bound for the KL divergence between the target and the estimated distributions, explicitly depending on any time-dependent noise schedule. Assuming that the score is Lipschitz continuous, we provide an improved error bound in Wasserstein distance, taking advantage of favourable underlying contraction mechanisms. We also propose an algorithm to automatically tune the noise schedule using the proposed upper bound. We illustrate empirically the performance of the noise schedule optimization in comparison to standard choices in the literature.

Constant (naive) imputation is still widely used in practice as this is a first easy-to-use technique to deal with missing data. Yet, this simple method could be expected to induce a large bias for prediction purposes, as the imputed input may strongly differ from the true underlying data. However, recent works suggest that this bias is low in the context of high-dimensional linear predictors when data is supposed to be missing completely at random (MCAR). This paper completes the picture for linear predictors by confirming the intuition that the bias is negligible and that surprisingly naive imputation also remains relevant in very low dimension.To this aim, we consider a unique underlying random features model, which offers a rigorous framework for studying predictive performances, whilst the dimension of the observed features varies.Building on these theoretical results, we establish finite-sample bounds on stochastic gradient (SGD) predictors applied to zero-imputed data, a strategy particularly well suited for large-scale learning.If the MCAR assumption appears to be strong, we show that similar favorable behaviors occur for more complex missing data scenarios.

Missing values are more and more present as the size of datasets increases. These missing values can occur for a variety of reasons, such as sensor failures, refusals to answer poll questions, or aggregations of data coming from different sources (with different methods of data collection). There may be different processes of missing value generation on the same dataset, which makes the task of data cleaning difficult or impossible without creating large biases. In his leading work, Rubin [1976] distinguishes three missing values scenarios: Missing Completely At Random (MCAR), Missing At Random (MAR), and Missing Not At Random (MNAR), depending on the links between the observed variables, the missing ones, and the missing pattern. In the linear regression framework, most of the literature focuses on parameter estimation [Little, 1992, Jones, 1996], using sometimes a sparse prior leading to the Lasso estimator [Loh and Wainwright, 2012] or the Dantzig selector [Rosenbaum and Tsybakov, 2010]. Note that the robust estimation literature [Dalalyan and Thompson, 2019, Chen and Caramanis, 2013] could be also used to handle missing values, as the latter can be reinterpreted as a multiplicative noise in linear models.

In recent decades, technological advances have made it possible to collect large data sets. In this context, the model-based clustering is a very popular, flexible and interpretable methodology for data exploration in a well-defined statistical framework. One of the ironies of the increase of large datasets is that missing values are more frequent. However, traditional ways (as discarding observations with missing values or imputation methods) are not designed for the clustering purpose. In addition, they rarely apply to the general case, though frequent in practice, of Missing Not At Random (MNAR) values, i.e. when the missingness depends on the unobserved data values and possibly on the observed data values. The goal of this paper is to propose a novel approach by embedding MNAR data directly within model-based clustering algorithms. We introduce a selection model for the joint distribution of data and missing-data indicator. It corresponds to a mixture model for the data distribution and a general MNAR model for the missing-data mechanism, which may depend on the underlying classes (unknown) and/or the values of the missing variables themselves. A large set of meaningful MNAR sub-models is derived and the identifiability of the parameters is studied for each of the sub-models, which is usually a key issue for any MNAR proposals. The EM and Stochastic EM algorithms are considered for estimation. Finally, we perform empirical evaluations for the proposed submodels on synthetic data and we illustrate the relevance of our method on a medical register, the TraumaBase (R) dataset.