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This paper studies linear reconstruction of partially observed functional data which are recorded on a discrete grid. We propose a novel estimation approach based on approximate factor models with increasing rank taking into account potential covariate information. Whereas alternative reconstruction procedures commonly involve some preliminary smoothing, our method separates the signal from noise and reconstructs missing fragments at once. We establish uniform convergence rates of our estimator and introduce a new method for constructing simultaneous prediction bands for the missing trajectories. A simulation study examines the performance of the proposed methods in finite samples. Finally, a real data application of temperature curves demonstrates that our theory provides a simple and effective method to recover missing fragments.

We revisit the general framework introduced by Fazylab et al. (SIAM J. Optim. 28, 2018) to construct Lyapunov functions for optimization algorithms in discrete and continuous time. For smooth, strongly convex objective functions, we relax the requirements necessary for such a construction. As a result we are able to prove for Polyak's ordinary differential equations and for a two-parameter family of Nesterov algorithms rates of convergence that improve on those available in the literature. We analyse the interpretation of Nesterov algorithms as discretizations of the Polyak equation. We show that the algorithms are instances of Additive Runge-Kutta integrators and discuss the reasons why most discretizations of the differential equation do not result in optimization algorithms with acceleration. We also introduce a modification of Polyak's equation and study its convergence properties. Finally we extend the general framework to the stochastic scenario and consider an application to random algorithms with acceleration for overparameterized models; again we are able to prove convergence rates that improve on those in the literature.

The expansion of streaming media and e-commerce has led to a boom in recommendation systems, including Sequential recommendation systems, which consider the user's previous interactions with items. In recent years, research has focused on architectural improvements such as transformer blocks and feature extraction that can augment model information. Among these features are context and attributes. Of particular importance is the temporal footprint, which is often considered part of the context and seen in previous publications as interchangeable with positional information. Other publications use positional encodings with little attention to them. In this paper, we analyse positional encodings, showing that they provide relative information between items that are not inferable from the temporal footprint. Furthermore, we evaluate different encodings and how they affect metrics and stability using Amazon datasets. We added some new encodings to help with these problems along the way. We found that we can reach new state-of-the-art results by finding the correct positional encoding, but more importantly, certain encodings stabilise the training.

Numerical modeling is essential for comprehending intricate physical phenomena in different domains. To handle complexity, sensitivity analysis, particularly screening, is crucial for identifying influential input parameters. Kernel-based methods, such as the Hilbert Schmidt Independence Criterion (HSIC), are valuable for analyzing dependencies between inputs and outputs. Moreover, due to the computational expense of such models, metamodels (or surrogate models) are often unavoidable. Implementing metamodels and HSIC requires data from the original model, which leads to the need for space-filling designs. While existing methods like Latin Hypercube Sampling (LHS) are effective for independent variables, incorporating dependence is challenging. This paper introduces a novel LHS variant, Quantization-based LHS, which leverages Voronoi vector quantization to address correlated inputs. The method ensures comprehensive coverage of stratified variables, enhancing distribution across marginals. The paper outlines expectation estimators based on Quantization-based LHS in various dependency settings, demonstrating their unbiasedness. The method is applied on several models of growing complexities, first on simple examples to illustrate the theory, then on more complex environmental hydrological models, when the dependence is known or not, and with more and more interactive processes and factors. The last application is on the digital twin of a French vineyard catchment (Beaujolais region) to design a vegetative filter strip and reduce water, sediment and pesticide transfers from the fields to the river. Quantization-based LHS is used to compute HSIC measures and independence tests, demonstrating its usefulness, especially in the context of complex models.

Class imbalance in real-world data poses a common bottleneck for machine learning tasks, since achieving good generalization on under-represented examples is often challenging. Mitigation strategies, such as under or oversampling the data depending on their abundances, are routinely proposed and tested empirically, but how they should adapt to the data statistics remains poorly understood. In this work, we determine exact analytical expressions of the generalization curves in the high-dimensional regime for linear classifiers (Support Vector Machines). We also provide a sharp prediction of the effects of under/oversampling strategies depending on class imbalance, first and second moments of the data, and the metrics of performance considered. We show that mixed strategies involving under and oversampling of data lead to performance improvement. Through numerical experiments, we show the relevance of our theoretical predictions on real datasets, on deeper architectures and with sampling strategies based on unsupervised probabilistic models.

We consider covariance parameter estimation for Gaussian processes with functional inputs. From an increasing-domain asymptotics perspective, we prove the asymptotic consistency and normality of the maximum likelihood estimator. We extend these theoretical guarantees to encompass scenarios accounting for approximation errors in the inputs, which allows robustness of practical implementations relying on conventional sampling methods or projections onto a functional basis. Loosely speaking, both consistency and normality hold when the approximation error becomes negligible, a condition that is often achieved as the number of samples or basis functions becomes large. These later asymptotic properties are illustrated through analytical examples, including one that covers the case of non-randomly perturbed grids, as well as several numerical illustrations.

Models of complex technological systems inherently contain interactions and dependencies among their input variables that affect their joint influence on the output. Such models are often computationally expensive and few sensitivity analysis methods can effectively process such complexities. Moreover, the sensitivity analysis field as a whole pays limited attention to the nature of interaction effects, whose understanding can prove to be critical for the design of safe and reliable systems. In this paper, we introduce and extensively test a simple binning approach for computing sensitivity indices and demonstrate how complementing it with the smart visualization method, simulation decomposition (SimDec), can permit important insights into the behavior of complex engineering models. The simple binning approach computes first-, second-order effects, and a combined sensitivity index, and is considerably more computationally efficient than the mainstream measure for Sobol indices introduced by Saltelli et al. The totality of the sensitivity analysis framework provides an efficient and intuitive way to analyze the behavior of complex systems containing interactions and dependencies.

Disaggregated evaluation is a central task in AI fairness assessment, where the goal is to measure an AI system's performance across different subgroups defined by combinations of demographic or other sensitive attributes. The standard approach is to stratify the evaluation data across subgroups and compute performance metrics separately for each group. However, even for moderately-sized evaluation datasets, sample sizes quickly get small once considering intersectional subgroups, which greatly limits the extent to which intersectional groups are included in analysis. In this work, we introduce a structured regression approach to disaggregated evaluation that we demonstrate can yield reliable system performance estimates even for very small subgroups. We provide corresponding inference strategies for constructing confidence intervals and explore how goodness-of-fit testing can yield insight into the structure of fairness-related harms experienced by intersectional groups. We evaluate our approach on two publicly available datasets, and several variants of semi-synthetic data. The results show that our method is considerably more accurate than the standard approach, especially for small subgroups, and demonstrate how goodness-of-fit testing helps identify the key factors that drive differences in performance.

In many modern regression applications, the response consists of multiple categorical random variables whose probability mass is a function of a common set of predictors. In this article, we propose a new method for modeling such a probability mass function in settings where the number of response variables, the number of categories per response, and the dimension of the predictor are large. Our method relies on a functional probability tensor decomposition: a decomposition of a tensor-valued function such that its range is a restricted set of low-rank probability tensors. This decomposition is motivated by the connection between the conditional independence of responses, or lack thereof, and their probability tensor rank. We show that the model implied by such a low-rank functional probability tensor decomposition can be interpreted in terms of a mixture of regressions and can thus be fit using maximum likelihood. We derive an efficient and scalable penalized expectation maximization algorithm to fit this model and examine its statistical properties. We demonstrate the encouraging performance of our method through both simulation studies and an application to modeling the functional classes of genes.

We consider optimal experimental design (OED) for nonlinear inverse problems within the Bayesian framework. Optimizing the data acquisition process for large-scale nonlinear Bayesian inverse problems is a computationally challenging task since the posterior is typically intractable and commonly-encountered optimality criteria depend on the observed data. Since these challenges are not present in OED for linear Bayesian inverse problems, we propose an approach based on first linearizing the associated forward problem and then optimizing the experimental design. Replacing an accurate but costly model with some linear surrogate, while justified for certain problems, can lead to incorrect posteriors and sub-optimal designs if model discrepancy is ignored. To avoid this, we use the Bayesian approximation error (BAE) approach to formulate an A-optimal design objective for sensor selection that is aware of the model error. In line with recent developments, we prove that this uncertainty-aware objective is independent of the exact choice of linearization. This key observation facilitates the formulation of an uncertainty-aware OED objective function using a completely trivial linear map, the zero map, as a surrogate to the forward dynamics. The base methodology is also extended to marginalized OED problems, accommodating uncertainties arising from both linear approximations and unknown auxiliary parameters. Our approach only requires parameter and data sample pairs, hence it is particularly well-suited for black box forward models. We demonstrate the effectiveness of our method for finding optimal designs in an idealized subsurface flow inverse problem and for tsunami detection.