The example shows the learned IT map with high w =2 0 approximating the ET map.

In many unpaired image domain translation problems, e.g., style transfer or super-resolution, it is important to keep the translated image similar to its respective input image.

Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose \texttt{FairSinkhorn}, a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalised OT problem, for which we derive novel finite-sample complexity guarantees. This result is of independent interest as it can be generalized to arbitrary convex penalties. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound guaranteeing that the learned cost yields fair matchings on unseen data. Finally, we present empirical results that illustrate the trade-offs between fairness and performance.

Topological data analysis (TDA) has been used successfully in a wide array of applications, for instance in medical (Nicolau et al., 2011) or material (Hiraoka et al., 2016) sciences, computer vision (Li et al., 2014) or to classify NBA players (Lum et al., 2013).

Topological data analysis (TDA) has been used successfully in a wide array of applications, for instance in medical (Nicolau et al., 2011) or material (Hiraoka et al., 2016) sciences, computer vision (Li et al., 2014) or to classify NBA players (Lum et al., 2013).

However, they tend to suffer a large computational burden.

Optimal transport (OT) has emerged as a fundamental computational tool across many areas, including machine learning, computer vision, statistics, and biology [Arjovsky et al., 2017, Peyr e and Cuturi, 2019, Schiebinger et al., 2019, Bonneel and Digne, 2023]. It provides a principled framework for comparing probability distributions, and it has a rich mathematical history [Villani et al., 2008]. While the combination of practical utility and deep mathematical theory has led to the broad adoption of OT ideas in mathematics, science, and engineering, finding ways to scale OT solutions and make them interpretable remains a fundamental research question [Cuturi et al., 2023, Khamis et al., 2024]. In particular, OT typically suffers from the curse of dimensionality [Chewi et al., 2025], and regularized estimators may lack sparsity [Genevay et al., 2019]. A long line of work has focused on making OT scalable and interpretable through regularization. The most classical of these is entropic regularization, which yields a strictly convex program that can be solved via Sinkhorn scaling [Sinkhorn, 1967, Cuturi, 2013]. More recent work has sought to increase efficiency and interpretability through quadratic regularization [Blondel et al., 2018, Lorenz et al., 2021], as well as low-rank factorizations [Forrow et al., 2019, Scetbon et al., 2021]. These methods show promise in biological applications, particularly in single-cell RNA sequencing analysis [Klein et al., 2025]. Another closely related set of recent works attempts to include sparsity in the OT map using elastic costs Cuturi et al. [2023], Klein et al. [2024], Chen et al. [2025].

The example shows the learned IT map with high w =2 0 approximating the ET map.

In many unpaired image domain translation problems, e.g., style transfer or super-resolution, it is important to keep the translated image similar to its respective input image.

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