arXiv subjects

Alberto González-Sanz

Publications and source records attributed to Alberto González-Sanz.

At least 37 records · Page 2Linked to original sources

arXiv2026

Sparsity of Quadratically Regularized Optimal Transport: Scalar Case

The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling $π_{\varepsilon}$ has sparse support for small regularization parameter $\varepsilon$, in contrast to entropic regularization whose solutions have full support for any $\varepsilon>0$. Focusing on continuous and scalar marginals, we provide the first precise description of this sparsity. Namely, we show that the support of $π_{\varepsilon}$ shrinks to the Monge graph at the sharp rate $\varepsilon^{1/3}$. This result is based on a detailed analysis of the dual potential $f_{\varepsilon}$ for small $\varepsilon$. In particular, we prove that $f_{\varepsilon}$ is twice differentiable a.s. and bound the second derivative uniformly in $\varepsilon$, showing that $f_{\varepsilon}$ is uniformly strongly convex. Convergence rates for $f_{\varepsilon}$ and its derivative are also obtained.

math.OC↗

arXiv2025

Linearization of Monge-Ampère Equations and Statistical Applications

Optimal transport has found numerous applications across data science, many of which require differentiating the optimal transport map with respect to the underlying probability densities in the Fréchet sense. In this work, we show that when the reference measure $Q$ is sufficiently regular in space and the curve of target measures $\{P_t\}_{t\in I}$ is both spatially regular and $\mathcal{C}^1$ in time, then the associated curve of optimal transport maps $\{\nabla ϕ_t\}_{t\in I}$ pushing $Q$ toward $P_t$ is itself a $\mathcal{C}^1$ curve. Moreover, we identify its time derivative as the solution to the \emph{linearized Monge--Ampère equation}, a second-order elliptic PDE with strictly oblique boundary conditions and a vanishing zero-order term. Our proof relies on applying the implicit function theorem to the Monge--Ampère equation with natural boundary conditions. As consequences, we establish regularity of the transport-based quantile regressor with respect to the covariates and derive a central limit theorem for smooth optimal transport maps.

math.AP↗

arXiv2025

Sample Complexity of Quadratically Regularized Optimal Transport

It is well known that optimal transport suffers from the curse of dimensionality: when the prescribed marginals are approximated by i.i.d. samples, the convergence of the empirical optimal transport problem to the population counterpart slows exponentially with increasing dimension. Entropically regularized optimal transport (EOT) has become the standard bearer in many statistical applications as it avoids this curse. Indeed, EOT has parametric sample complexity, as has been shown in a series of works based on the smoothness of the EOT potentials or the strong concavity of the dual EOT problem. However, EOT produces full-support approximations to the (sparse) OT problem, leading to overspreading in applications, and is computationally unstable for small regularization parameters. The most popular alternative is quadratically regularized optimal transport (QOT), which penalizes couplings by $L^2$ norm instead of relative entropy. QOT produces sparse approximations of OT and is computationally stable. However, its potentials are not smooth (do not belong to a Donsker class) and its dual problem is not strongly concave, hence QOT is often assumed to suffer from the curse of dimensionality. In this paper, we show that QOT nevertheless has parametric sample complexity. More precisely, we establish central limit theorems for its dual potentials, optimal couplings, and optimal costs. Our analysis is based on novel arguments that focus on the regularity of the support of the optimal QOT coupling. Specifically, we establish a Lipschitz property of its sections and leverage VC theory to bound its statistical complexity. Our analysis also leads to gradient estimates of independent interest, including $C^{1,1}$ regularity of the population potentials.

math.ST↗

arXiv2025

On the breakdown point of transport-based quantiles

Recent work has used optimal transport ideas to generalize the notion of (center-outward) quantiles to dimension $d\geq 2$. We study the robustness properties of these transport-based quantiles by deriving their breakdown point, roughly, the smallest amount of contamination required to make these quantiles take arbitrarily aberrant values. We prove that the transport median defined in Chernozhukov et al.~(2017) and Hallin et al.~(2021) has breakdown point of $1/2$. Moreover, a point in the transport depth contour of order $τ\in [0,1/2]$ has breakdown point of $τ$. This shows that the multivariate transport depth shares the same breakdown properties as its univariate counterpart. Our proof relies on a general argument connecting the breakdown point of transport maps evaluated at a point to the Tukey depth of that point in the reference measure.

math.ST↗

arXiv2025

Nonparametric Vector Quantile Autoregression

Prediction is a key issue in time series analysis. Just as classical mean regression models, classical autoregressive methods, yielding L$^2$ point-predictions, provide rather poor predictive summaries; a much more informative approach is based on quantile (auto)regression, where the whole distribution of future observations conditional on the past is consistently recovered. Since their introduction by Koenker and Xiao in 2006, autoregressive quantile autoregression methods have become a popular and successful alternative to the traditional L$^2$ ones. Due to the lack of a widely accepted concept of multivariate quantiles, however, quantile autoregression methods so far have been limited to univariate time series. Building upon recent measure-transportation-based concepts of multivariate quantiles, we develop here a nonparametric vector quantile autoregressive approach to the analysis and prediction of (nonlinear as well as linear) multivariate time series.

math.ST↗

arXiv2025

Improved learning theory for kernel distribution regression with two-stage sampling

The distribution regression problem encompasses many important statistics and machine learning tasks, and arises in a large range of applications. Among various existing approaches to tackle this problem, kernel methods have become a method of choice. Indeed, kernel distribution regression is both computationally favorable, and supported by a recent learning theory. This theory also tackles the two-stage sampling setting, where only samples from the input distributions are available. In this paper, we improve the learning theory of kernel distribution regression. We address kernels based on Hilbertian embeddings, that encompass most, if not all, of the existing approaches. We introduce the novel near-unbiased condition on the Hilbertian embeddings, that enables us to provide new error bounds on the effect of the two-stage sampling, thanks to a new analysis. We show that this near-unbiased condition holds for three important classes of kernels, based on optimal transport and mean embedding. As a consequence, we strictly improve the existing convergence rates for these kernels. Our setting and results are illustrated by numerical experiments.

math.ST↗

arXiv2025

Estimation of Algebraic Sets: Extending PCA Beyond Linearity

An algebraic set is defined as the zero locus of a system of real polynomial equations. In this paper we address the problem of recovering an unknown algebraic set $\mathcal{A}$ from noisy observations of latent points lying on $\mathcal{A}$ -- a task that extends principal component analysis, which corresponds to the purely linear case. Our procedure consists of three steps: (i) constructing the {\it moment matrix} from the Vandermonde matrix associated with the data set and the degree of the fitted polynomials, (ii) debiasing this moment matrix to remove the noise-induced bias, (iii) extracting its kernel via an eigenvalue decomposition of the debiased moment matrix. These steps yield $n^{-1/2}$-consistent estimators of the coefficients of a set of generators for the ideal of polynomials vanishing on $\mathcal{A}$. To reconstruct $\mathcal{A}$ itself, we propose three complementary strategies: (a) compute the zero set of the fitted polynomials; (b) build a semi-algebraic approximation that encloses $\mathcal{A}$; (c) when structural prior information is available, project the estimated coefficients onto the corresponding constrained space. We prove (nearly) parametric asymptotic error bounds and show that each approach recovers $\mathcal{A}$ under mild regularity conditions.

math.ST↗

arXiv2025

Stability of Mean-Field Variational Inference

Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. This paper studies the stability properties of the mean-field approximation when the target distribution varies within the class of strongly log-concave measures. We establish dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in the 2-Wasserstein distance, with Lipschitz constant inversely proportional to the log-concavity parameter. Under additional regularity conditions, we further show that the MFVI optimizer depends differentiably on the target potential and characterize the derivative by a partial differential equation. Methodologically, we follow a novel approach to MFVI via linearized optimal transport: the non-convex MFVI problem is lifted to a convex optimization over transport maps with a fixed base measure, enabling the use of calculus of variations and functional analysis. We discuss several applications of our results to robust Bayesian inference and empirical Bayes, including a quantitative Bernstein--von Mises theorem for MFVI, as well as to distributed stochastic control.

math.PR↗

arXiv2025

On the Nonconvexity of Push-Forward Constraints and Its Consequences in Machine Learning

The push-forward operation enables one to redistribute a probability measure through a deterministic map. It plays a key role in statistics and optimization: many learning problems (notably from optimal transport, generative modeling, and algorithmic fairness) include constraints or penalties framed as push-forward conditions on the model. However, the literature lacks general theoretical insights on the (non)convexity of such constraints and its consequences on the associated learning problems. This paper aims at filling this gap. In the first part, we provide a range of sufficient and necessary conditions for the (non)convexity of two sets of functions: the maps transporting one probability measure to another and the maps inducing equal output distributions across distinct probability measures. This highlights that for most probability measures, these push-forward constraints are not convex. In the second part, we show how this result implies critical limitations on the design of convex optimization problems for learning generative models or groupwise fair predictors. This work will hopefully help researchers and practitioners have a better understanding of the critical impact of push-forward conditions onto convexity.

stat.ML↗

arXiv2025

Sparse Regularized Optimal Transport without Curse of Dimensionality

Entropic optimal transport -- the optimal transport problem regularized by KL diver\-gence -- is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of dimensionality suffered by unregularized optimal transport. The flip side of smoothness is overspreading: the entropic coupling always has full support, whereas the unregularized coupling that it approximates is usually sparse, even given by a map. Regularizing optimal transport by less-smooth $f$-divergences such as Tsallis divergence (i.e., $L^p$-regularization) is known to allow for sparse approximations, but is often thought to suffer from the curse of dimensionality as the couplings have limited differentiability and the dual is not strongly concave. We refute this conventional wisdom and show, for a broad family of divergences, that the key empirical quantities converge at the parametric rate, independently of the dimension. More precisely, we provide central limit theorems for the optimal cost, the optimal coupling, and the dual potentials induced by i.i.d.\ samples from the marginals. These results are obtained by a powerful yet elementary approach that is of broader interest for Z-estimation in function classes that are not Donsker.

math.ST↗

arXiv2025

Quantitative Convergence of Quadratically Regularized Linear Programs

Linear programs with quadratic regularization are attracting renewed interest due to their applications in optimal transport: unlike entropic regularization, the squared-norm penalty gives rise to sparse approximations of optimal transport couplings. It is well known that the solution of a quadratically regularized linear program over any polytope converges stationarily to the minimal-norm solution of the linear program when the regularization parameter tends to zero. However, that result is merely qualitative. Our main result quantifies the convergence by specifying the exact threshold for the regularization parameter, after which the regularized solution also solves the linear program. Moreover, we bound the suboptimality of the regularized solution before the threshold. These results are complemented by a convergence rate for the regime of large regularization. We apply our general results to the setting of optimal transport, where we shed light on how the threshold and suboptimality depend on the number of data points.

math.OC↗

arXiv2025

Monotonicity in Quadratically Regularized Linear Programs

In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the unregularized counterpart as the regularization parameter is relaxed. We find it useful to investigate this monotonicity more abstractly for linear programs over polytopes, regularized with the squared norm. Here, monotonicity can be stated as an invariance property of the curve mapping the regularization parameter to the solution: once the curve enters a face of the polytope, does it remain in that face forever? We show that this invariance is equivalent to a geometric property of the polytope, namely that each face contains the minimum norm point of its affine hull. Returning to the optimal transport problem and its associated Birkhoff polytope, we verify this property for low dimensions, but show that it fails for marginals with five or more point masses. As a consequence, the conjectured monotonicity of the support fails in general, even if experiments suggest that monotonicity holds for many cost matrices. Separately, we apply our geometric point of view to a problem of Erdős, namely to characterize the doubly stochastic matrices whose maximal trace equals their squared norm.

math.OC↗

arXiv2024

Infinitesimal behavior of Quadratically Regularized Optimal Transport and its relation with the Porous Medium Equation

The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be \emph{sparse}, i.e., the density of the coupling needs to be zero for a large subset of the product of the supports of the marginals. However, unlike the acclaimed entropy-regularized optimal transport, the effect of quadratic regularization on the transport problem is not well understood from a mathematical standpoint. In this work, we take a first step towards its understanding. We prove that the difference between the cost of optimal transport and its regularized version multiplied by the ratio $\varepsilon^{-\frac{2}{d+2}}$ converges to a nontrivial limit as the regularization parameter $\varepsilon$ tends to 0. The proof confirms a conjecture from Zhang et al. (2023) where it is claimed that a modification of the self-similar solution of the porous medium equation, the Barenblatt--Pattle solution, can be used as an approximate solution of the regularized transport cost for small values of $\varepsilon$.

math.AP↗

arXiv2024

Nonlinear Inverse Optimal Transport: Identifiability of the Transport Cost from its Marginals and Optimal Values

The inverse optimal transport problem is to find the underlying cost function from the knowledge of optimal transport plans. While this amounts to solving a linear inverse problem, in this work we will be concerned with the nonlinear inverse problem to identify the cost function when only a set of marginals and its corresponding optimal values are given. We focus on absolutely continuous probability distributions with respect to the $d$-dimensional Lebesgue measure and classes of concave and convex cost functions. Our main result implies that the cost function is uniquely determined from the union of the ranges of the gradients of the optimal potentials. Since, in general, the optimal potentials may not be observed, we derive sufficient conditions for their identifiability - if an open set of marginals is observed, the optimal potentials are then identified via the value of the optimal costs. We conclude with a more in-depth study of this problem in the univariate case, where an explicit representation of the transport plan is available. Here, we link the notion of identifiability of the cost function with that of statistical completeness.

math.OC↗

arXiv2023

Two-sample goodness-of-fit tests on the flat torus based on Wasserstein distance and their relevance to structural biology

This work is motivated by the study of local protein structure, which is defined by two variable dihedral angles that take values from probability distributions on the flat torus. Our goal is to provide the space $\mathcal{P}(\mathbb{R}^2/\mathbb{Z}^2)$ with a metric that quantifies local structural modifications due to changes in the protein sequence, and to define associated two-sample goodness-of-fit testing approaches. Due to its adaptability to the space geometry, we focus on the Wasserstein distance as a metric between distributions. We extend existing results of the theory of Optimal Transport to the $d$-dimensional flat torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$, in particular a Central Limit Theorem. Moreover, we propose different approaches for two-sample goodness-of-fit testing for the one and two-dimensional case, based on the Wasserstein distance. We prove their validity and consistency. We provide an implementation of these tests in \textsf{R}. Their performance is assessed by numerical experiments on synthetic data and illustrated by an application to protein structure data.

stat.ME↗

arXiv2023

Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability

The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann's celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures $\rm P$ and $\rm Q$ on a separable Hilbert space $\mathcal{H}$ where $\rm P$ does not give mass to "small sets" (namely, Lipschitz hypersurfaces), we show, without imposing any moment assumptions, that there exists a gradient of convex function $\nablaψ$ pushing ${\rm P} $ forward to ${\rm Q}$. In case $\mathcal{H}$ is infinite-dimensional, ${\rm P}$-a.s. uniqueness is not guaranteed, though. If, however, ${\rm Q}$ is boundedly supported (a natural assumption in several statistical applications), then this gradient is ${\rm P}$ a.s. unique. In the second part of the paper, we establish stability results for transport maps in the sense of uniform convergence over compact "regularity sets". As a consequence, we obtain a central limit theorem for the fluctuations of the optimal quadratic transport cost in a separable Hilbert space.

math.PR↗

arXiv2023

Weak Limits for Empirical Entropic Optimal Transport: Beyond Smooth Costs

We establish weak limits for the empirical entropy regularized optimal transport cost, the expectation of the empirical plan and the conditional expectation. Our results require only uniform boundedness of the cost function and no smoothness properties, thus emphasizing the far-reaching regularizing nature of entropy penalization. To derive these results, we employ a novel technique that sidesteps the intricacies linked to empirical process theory and the control of suprema of function classes determined by the cost. Instead, we perform a careful linearization analysis for entropic optimal transport with respect to an empirical $L^2$-norm, which enables a streamlined analysis. As a consequence, our work gives rise to new implications for a multitude of transport-based applications under general costs, including pointwise distributional limits for the empirical entropic optimal transport map estimator, kernel methods as well as regularized colocalization curves. Overall, our research lays the foundation for an expanded framework of statistical inference with empirical entropic optimal transport.

math.ST↗

arXiv2023

Transport-based Counterfactual Models

Counterfactual frameworks have grown popular in machine learning for both explaining algorithmic decisions but also defining individual notions of fairness, more intuitive than typical group fairness conditions. However, state-of-the-art models to compute counterfactuals are either unrealistic or unfeasible. In particular, while Pearl's causal inference provides appealing rules to calculate counterfactuals, it relies on a model that is unknown and hard to discover in practice. We address the problem of designing realistic and feasible counterfactuals in the absence of a causal model. We define transport-based counterfactual models as collections of joint probability distributions between observable distributions, and show their connection to causal counterfactuals. More specifically, we argue that optimal-transport theory defines relevant transport-based counterfactual models, as they are numerically feasible, statistically-faithful, and can coincide under some assumptions with causal counterfactual models. Finally, these models make counterfactual approaches to fairness feasible, and we illustrate their practicality and efficiency on fair learning. With this paper, we aim at laying out the theoretical foundations for a new, implementable approach to counterfactual thinking.

cs.AI↗