We study the estimation problem of distribution-on-distribution regression, where both predictors and responses are probability measures. Existing approaches typically rely on a global optimal transport map or tangent-space linearization, which can be restrictive in approximation capacity and distort geometry in multivariate underlying domains. In this paper, we propose the \emph{Neural Local Wasserstein Regression}, a flexible nonparametric framework that models regression through locally defined transport maps in Wasserstein space. Our method builds on the analogy with classical kernel regression: kernel weights based on the 2-Wasserstein distance localize estimators around reference measures, while neural networks parameterize transport operators that adapt flexibly to complex data geometries. This localized perspective broadens the class of admissible transformations and avoids the limitations of global map assumptions and linearization structures. We develop a practical training procedure using DeepSets-style architectures and Sinkhorn-approximated losses, combined with a greedy reference selection strategy for scalability. Through synthetic experiments on Gaussian and mixture models, as well as distributional prediction tasks on MNIST, we demonstrate that our approach effectively captures nonlinear and high-dimensional distributional relationships that elude existing methods.

In risk-averse reinforcement learning (RL), the goal is to optimize some risk measure of the returns.

Clustering is a central problem in unsupervised learning.

A naive implementation of this method requires two steps of communication per round to implement this algorithm.

A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools.

In Section A.1 and A.2 we derive closed-form expressions of the standard and robust risks from We first prove Equation (13). We now prove the second part of the statement. In this section we provide additional details on our experiments. B.1 Neural networks on sanitized binary MNIST If not mentioned otherwise, we use noiseless i.i.d. C.1 we give an intuitive explantion for the robust overfitting phenomenon described in C.2 we discuss how inconsistent adversarial training prevents We now shed light on the phenomena revealed by Theorem 3.1 and Figure 2. In particular, we In this section we further discuss robust logistic regression studied in Section 4. As observed in Section 4.4, label noise can prevent interpolation and hence improve the robust risk Hence, inconsistent training perturbations can induce spurious regularization effects.

Work done as an intern at the Gatsby Computational Neuroscience Unit, University College London 38th Conference on Neural Information Processing Systems (NeurIPS 2024).