# Distribution Regression with Sliced Wasserstein Kernels arXiv: https://arxiv.org/abs/2202.03926 ยท HTML: https://mlsanity.com/paper/2202.03926/ ## Key facts - Cited by 18 papers in the map (data through 14 Oct 2026). - Builds on 22 of its 64 references that are in the map. - First posted on arXiv 8 Feb 2022. ## Builds on (most cited first, 22 of 22) - [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017) - [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016) - [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019) - [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019) - [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019) - [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018) - [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020) - [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020) - [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017) - [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015) - [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014) - [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014) - [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019) - [Distributed learning with regularized least squares](https://mlsanity.com/paper/1608.03339/) (2016) - [The Advantage of Conditional Meta-Learning for Biased Regularization and Fine-Tuning](https://mlsanity.com/paper/2008.10857/) (2020) - [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020) - [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019) - [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013) - [A Gaussian Process Regression Model for Distribution Inputs](https://mlsanity.com/paper/1701.09055/) (2017) - [Robust Kernel-based Distribution Regression](https://mlsanity.com/paper/2104.10637/) (2021) - [Fast Distribution To Real Regression](https://mlsanity.com/paper/1311.2236/) (2013) - [Stochastic Gradient Descent Meets Distribution Regression](https://mlsanity.com/paper/2010.12842/) (2020) ## Built on by (most cited first, 18 of 18) - [Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals](https://mlsanity.com/paper/2303.05798/) (2023) - [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022) - [Statistical Optimality of Divide and Conquer Kernel-based Functional Linear Regression](https://mlsanity.com/paper/2211.10968/) (2022) - [Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates](https://mlsanity.com/paper/2402.01493/) (2024) - [Gaussian process regression with Sliced Wasserstein Weisfeiler-Lehman graph kernels](https://mlsanity.com/paper/2402.03838/) (2024) - [Gaussian Processes on Distributions based on Regularized Optimal Transport](https://mlsanity.com/paper/2210.06574/) (2022) - [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024) - [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024) - [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025) - [Fast and Scalable Score-Based Kernel Calibration Tests](https://mlsanity.com/paper/2510.14711/) (2025) - [Improved learning theory for kernel distribution regression with two-stage sampling](https://mlsanity.com/paper/2308.14335/) (2023) - [Distributional encoding for Gaussian process regression with qualitative inputs](https://mlsanity.com/paper/2506.04813/) (2025) - [Kernel-Based Evaluation of Conditional Biological Sequence Models](https://mlsanity.com/paper/2510.15601/) (2025) - [Bayesian Additive Distribution Regression](https://mlsanity.com/paper/2603.06462/) (2026) - [A reproducible comparative study of categorical kernels for Gaussian process regression, with new clustering-based nested kernels](https://mlsanity.com/paper/2510.01840/) (2025) - [Wasserstein Regression as a Variational Approximation of Probabilistic Trajectories through the Bernstein Basis](https://mlsanity.com/paper/2510.26607/) (2025) - [Statistical Learning Theory for Distributional Classification](https://mlsanity.com/paper/2601.14818/) (2026) - [Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty](https://mlsanity.com/paper/2603.17271/) (2026) ## Abstract (arXiv) The problem of learning functions over spaces of probabilities - or distribution regression - is gaining significant interest in the machine learning community. A key challenge behind this problem is to identify a suitable representation capturing all relevant properties of the underlying functional mapping. A principled approach to distribution regression is provided by kernel mean embeddings, which lifts kernel-induced similarity on the input domain at the probability level. This strategy effectively tackles the two-stage sampling nature of the problem, enabling one to derive estimators with strong statistical guarantees, such as universal consistency and excess risk bounds. However, kernel mean embeddings implicitly hinge on the maximum mean discrepancy (MMD), a metric on probabilities, which may fail to capture key geometrical relations between distributions. In contrast, optimal transport (OT) metrics, are potentially more appealing. In this work, we propose an OT-based estimator for distribution regression. We build on the Sliced Wasserstein distance to obtain an OT-based representation. We study the theoretical properties of a kernel ridge regression estimator based on such representation, for which we prove universal consistency and excess risk bounds. Preliminary experiments complement our theoretical findings by showing the effectiveness of the proposed approach and compare it with MMD-based estimators.