arXiv subjects

Michel Groppe

Publications and source records attributed to Michel Groppe.

4 recordsLinked to original sources

arXiv2025

Optimal Transport Based Testing in Factorial Designs

We introduce a general framework for testing statistical hypotheses in factorial designs for probability measures supported on finite spaces. The suggested methodology is based on the pairwise comparison of measures using optimal transport (OT). The formulation of hypotheses is intuitive: It is a direct extension of those underlying the analysis of variance (ANOVA) and its nonparametric counterparts to test for linear relationships between (discrete) probability measures in factorial designs. To this end, means or cumulative distribution functions simply will be replaced by measures. We derive under the null hypotheses and under (local) alternatives the asymptotic distribution of the corresponding empirical OT test statistic, which is the optimal value of a linear program with random objective function. It turns out that this requires to extend existing techniques from probability measures to signed measures, and we show directional Hadamard differentiability and the validity of the functional delta method. We discuss computational issues, permutation and bootstrap tests, and back up our findings with simulations. We illustrate our methodology on datasets from cellular biophysics and from biometric identification.

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arXiv2024

Identifiability and Exact Reconstruction of the Optimal Transport Cost on Finite Spaces

The goal of optimal transport (OT) is to find optimal assignments or matchings between data sets which minimize the total cost for a given cost function. However, sometimes the cost function is unknown but we have access to (parts of) the solution to the OT problem, e.g.\ the OT plan or the value of the objective function. Recovering the cost from such information is called inverse OT and has become recently of certain interest triggered by novel applications, e.g.\ in social science and economics. This raises the issue under which circumstances such cost is identifiable, i.e., it can be uniquely recovered from other OT quantities. In this work we provide sufficient and necessary conditions for the identifiability of the cost function on finite ground spaces. We find that such conditions correspond to the combinatorial structure of the corresponding linear program and discuss its computational complexity and implications for cost estimation in statistical linear models.

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arXiv2023

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.

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arXiv2023

Lower Complexity Adaptation for Empirical Entropic Optimal Transport

Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical bounds for empirical plug-in estimators of the EOT cost and show that their statistical performance in the entropy regularization parameter $\epsilon$ and the sample size $n$ only depends on the simpler of the two probability measures. For instance, under sufficiently smooth costs this yields the parametric rate $n^{-1/2}$ with factor $\epsilon^{-d/2}$, where $d$ is the minimum dimension of the two population measures. This confirms that empirical EOT also adheres to the lower complexity adaptation principle, a hallmark feature only recently identified for unregularized OT. As a consequence of our theory, we show that the empirical entropic Gromov-Wasserstein distance and its unregularized version for measures on Euclidean spaces also obey this principle. Additionally, we comment on computational aspects and complement our findings with Monte Carlo simulations. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class. Crucial to our analysis is the observation that the entropic cost-transformation of a function class does not increase its uniform metric entropy by much.

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