# E-ROBOT: a dimension-free method for robust statistics and machine learning via Schr"odinger bridge

arXiv: https://arxiv.org/abs/2509.11532 · HTML: https://mlsanity.com/paper/2509.11532/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 55 references that are in the map.
- First posted on arXiv 15 Sep 2025.

## Builds on (most cited first, 21 of 21)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis](https://mlsanity.com/paper/2111.01361/) (2021)
- [Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications](https://mlsanity.com/paper/2101.04039/) (2021)
- [Multivariate goodness-of-Fit tests based on Wasserstein distance](https://mlsanity.com/paper/2003.06684/) (2020)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy](https://mlsanity.com/paper/2306.06940/) (2023)
- [Quantiles and Quantile Regression on Riemannian Manifolds: a measure-transportation-based approach](https://mlsanity.com/paper/2410.15711/) (2024)
- [Saddlepoint approximations for spatial panel data models](https://mlsanity.com/paper/2001.10377/) (2020)
- [On the use of the cumulant generating function for inference on time series](https://mlsanity.com/paper/2403.12714/) (2024)

## Abstract (arXiv)

We propose the Entropic-regularized Robust Optimal Transport (E-ROBOT) framework, a novel method that combines the robustness of ROBOT with the computational and statistical benefits of entropic regularization. We show that, rooted in the Schr"odinger bridge problem theory, E-ROBOT defines the robust Sinkhorn divergence overlineWvarepsilon,λ, where the parameter λ controls robustness and varepsilon governs the regularization strength. Letting n∈ mathbbN denote the sample size, a central theoretical contribution is establishing that the sample complexity of overlineWvarepsilon,λ is O(n-1/2), thereby avoiding the curse of dimensionality that plagues standard ROBOT. This dimension-free property unlocks the use of overlineWvarepsilon,λ as a loss function in large-dimensional statistical and machine learning tasks. With this regard, we demonstrate its utility through four applications: goodness-of-fit testing; computation of barycenters for corrupted 2D and 3D shapes; definition of gradient flows; and image colour transfer. From the computation standpoint, a perk of our novel method is that it can be easily implemented by modifying existing (textttPython) routines. From the theoretical standpoint, our work opens the door to many research directions in statistics and machine learning: we discuss some of them.