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1. Outlier-Robust Optimal Transport

   Mukherjee, Debarghya; Guha, Aritra; Solomon, Justin; Sun, Yuekai; Yurochkin, Mikhail (April 2021, International Conference on Machine Learning)

   Optimal transport (OT) measures distances between distributions in a way that depends on the geometry of the sample space. In light of recent advances in computational OT, OT distances are widely used as loss functions in machine learning. Despite their prevalence and advantages, OT loss functions can be extremely sensitive to outliers. In fact, a single adversarially-picked outlier can increase the standard W2-distance arbitrarily. To address this issue, we propose an outlier-robust formulation of OT. Our formulation is convex but challenging to scale at a first glance. Our main contribution is deriving an \emph{equivalent} formulation based on cost truncation that is easy to incorporate into modern algorithms for computational OT. We demonstrate the benefits of our formulation in mean estimation problems under the Huber contamination model in simulations and outlier detection tasks on real data. 

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2. Statistical inference with regularized optimal transport

   https://doi.org/10.1093/imaiai/iaad056  

   Goldfeld, Ziv; Kato, Kengo; Rioux, Gabriel; Sadhu, Ritwik (February 2024, Information and Inference: A Journal of the IMA)

   Abstract Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of (i) average- and max-sliced $$p$$-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples. 

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3. Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation

   Balaji, Yogesh; Chellappa, Rama; Feizi, Soheil (January 2020, Advances in Neural Information Processing Systems Foundation (NeurIPS))

   null (Ed.)

   Optimal Transport (OT) distances such as Wasserstein have been used in several areas such as GANs and domain adaptation. OT, however, is very sensitive to outliers (samples with large noise) in the data since in its objective function, every sample, including outliers, is weighed similarly due to the marginal constraints. To remedy this issue, robust formulations of OT with unbalanced marginal constraints have previously been proposed. However, employing these methods in deep learning problems such as GANs and domain adaptation is challenging due to the instability of their dual optimization solvers. In this paper, we resolve these issues by deriving a computationally-efficient dual form of the robust OT optimization that is amenable to modern deep learning applications. We demonstrate the effectiveness of our formulation in two applications of GANs and domain adaptation. Our approach can train state-of-the-art GAN models on noisy datasets corrupted with outlier distributions. In particular, our optimization computes weights for training samples reflecting how difficult it is for those samples to be generated in the model. In domain adaptation, our robust OT formulation leads to improved accuracy compared to the standard adversarial adaptation methods. 

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4. Projection‐based techniques for high‐dimensional optimal transport problems

   https://doi.org/10.1002/wics.1587  

   Zhang, Jingyi; Ma, Ping; Zhong, Wenxuan; Meng, Cheng (May 2022, WIREs Computational Statistics)

   Abstract Optimal transport (OT) methods seek a transformation map (or plan) between two probability measures, such that the transformation has the minimum transportation cost. Such a minimum transport cost, with a certain power transform, is called the Wasserstein distance. Recently, OT methods have drawn great attention in statistics, machine learning, and computer science, especially in deep generative neural networks. Despite its broad applications, the estimation of high‐dimensional Wasserstein distances is a well‐known challenging problem owing to the curse‐of‐dimensionality. There are some cutting‐edge projection‐based techniques that tackle high‐dimensional OT problems. Three major approaches of such techniques are introduced, respectively, the slicing approach, the iterative projection approach, and the projection robust OT approach. Open challenges are discussed at the end of the review. This article is categorized under:Statistical and Graphical Methods of Data Analysis > Dimension ReductionStatistical Learning and Exploratory Methods of the Data Sciences > Manifold Learning 

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5. Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark

   Korotin, Alexander; Li, Lingxiao; Genevay, Aude; Solomon, Justin; Filippov, Alexander; Burnaev, Evgeny (December 2021, NeurIPS)

   Despite the recent popularity of neural network-based solvers for optimal transport (OT), there is no standard quantitative way to evaluate their performance. In this paper, we address this issue for quadratic-cost transport---specifically, computation of the Wasserstein-2 distance, a commonly-used formulation of optimal transport in machine learning. To overcome the challenge of computing ground truth transport maps between continuous measures needed to assess these solvers, we use input-convex neural networks (ICNN) to construct pairs of measures whose ground truth OT maps can be obtained analytically. This strategy yields pairs of continuous benchmark measures in high-dimensional spaces such as spaces of images. We thoroughly evaluate existing optimal transport solvers using these benchmark measures. Even though these solvers perform well in downstream tasks, many do not faithfully recover optimal transport maps. To investigate the cause of this discrepancy, we further test the solvers in a setting of image generation. Our study reveals crucial limitations of existing solvers and shows that increased OT accuracy does not necessarily correlate to better results downstream. 

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