# Robust Kernel-based Distribution Regression

arXiv: https://arxiv.org/abs/2104.10637 · HTML: https://mlsanity.com/paper/2104.10637/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 21 Apr 2021.

## Builds on (most cited first, 8 of 8)

- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013)
- [Consistency Analysis of an Empirical Minimum Error Entropy Algorithm](https://mlsanity.com/paper/1412.5272/) (2014)
- [A Statistical Learning Approach to Modal Regression](https://mlsanity.com/paper/1702.05960/) (2017)
- [Stochastic Gradient Descent Meets Distribution Regression](https://mlsanity.com/paper/2010.12842/) (2020)
- [Universal Consistency and Robustness of Localized Support Vector Machines](https://mlsanity.com/paper/1703.06528/) (2017)
- [Learning Theory of Distributed Regression with Bias Corrected Regularization Kernel Network](https://mlsanity.com/paper/1708.01960/) (2017)
- [A Statistical Learning Assessment of Huber Regression](https://mlsanity.com/paper/2009.12755/) (2020)

## Built on by (most cited first, 7 of 7)

- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Learning Theory of Distribution Regression with Neural Networks](https://mlsanity.com/paper/2307.03487/) (2023)
- [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024)
- [Coefficient-based Regularized Distribution Regression](https://mlsanity.com/paper/2208.12427/) (2022)
- [Distributed Gradient Descent for Functional Learning](https://mlsanity.com/paper/2305.07408/) (2023)
- [Theory of Decentralized Robust Kernel-Based Learning](https://mlsanity.com/paper/2506.05215/) (2025)
- [Generalization Analysis of Transformers in Distribution Regression](https://mlsanity.com/paper/2606.29256/) (2026)

## Abstract (arXiv)

Regularization schemes for regression have been widely studied in learning theory and inverse problems. In this paper, we study distribution regression (DR) which involves two stages of sampling, and aims at regressing from probability measures to real-valued responses over a reproducing kernel Hilbert space (RKHS). Recently, theoretical analysis on DR has been carried out via kernel ridge regression and several learning behaviors have been observed. However, the topic has not been explored and understood beyond the least square based DR. By introducing a robust loss function lσ for two-stage sampling problems, we present a novel robust distribution regression (RDR) scheme. With a windowing function V and a scaling parameter σ which can be appropriately chosen, lσ can include a wide range of popular used loss functions that enrich the theme of DR. Moreover, the loss lσ is not necessarily convex, hence largely improving the former regression class (least square) in the literature of DR. The learning rates under different regularity ranges of the regression function fρ are comprehensively studied and derived via integral operator techniques. The scaling parameter σ is shown to be crucial in providing robustness and satisfactory learning rates of RDR.