# Improved learning theory for kernel distribution regression with two-stage sampling

arXiv: https://arxiv.org/abs/2308.14335 · HTML: https://mlsanity.com/paper/2308.14335/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 52 references that are in the map.
- First posted on arXiv 28 Aug 2023.

## Builds on (most cited first, 16 of 16)

- [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)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [A Gaussian Process Regression Model for Distribution Inputs](https://mlsanity.com/paper/1701.09055/) (2017)
- [Gaussian Processes on Distributions based on Regularized Optimal Transport](https://mlsanity.com/paper/2210.06574/) (2022)
- [Fast Distribution To Real Regression](https://mlsanity.com/paper/1311.2236/) (2013)
- [Understanding the 2016 US Presidential Election using ecological inference and distribution regression with census microdata](https://mlsanity.com/paper/1611.03787/) (2016)

## Built on by (most cited first, 1 of 1)

- [Bayesian Additive Distribution Regression](https://mlsanity.com/paper/2603.06462/) (2026)

## Abstract (arXiv)

The distribution regression problem encompasses many important statistics and machine learning tasks, and arises in a large range of applications. Among various existing approaches to tackle this problem, kernel methods have become a method of choice. Indeed, kernel distribution regression is both computationally favorable, and supported by a recent learning theory. This theory also tackles the two-stage sampling setting, where only samples from the input distributions are available. In this paper, we improve the learning theory of kernel distribution regression. We address kernels based on Hilbertian embeddings, that encompass most, if not all, of the existing approaches. We introduce the novel near-unbiased condition on the Hilbertian embeddings, that enables us to provide new error bounds on the effect of the two-stage sampling, thanks to a new analysis. We show that this near-unbiased condition holds for three important classes of kernels, based on optimal transport and mean embedding. As a consequence, we strictly improve the existing convergence rates for these kernels. Our setting and results are illustrated by numerical experiments.