# Convolutional Maximum Mean Discrepancy for Inference in Noisy Data

arXiv: https://arxiv.org/abs/2604.12022 · HTML: https://mlsanity.com/paper/2604.12022/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 81 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [Denoising Data with Measurement Error Using a Reproducing Kernel-based Diffusion Model](https://mlsanity.com/paper/2501.00212/) (2024)
- [Total robustness in Bayesian Nonlinear Regression](https://mlsanity.com/paper/2510.03131/) (2025)

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

- [Nonparametric Deconvolution and Denoising using Simulation Based Inference](https://mlsanity.com/paper/2606.21907/) (2026)

## Abstract (arXiv)

Modern data analyses frequently encounter settings where samples of variables are contaminated by measurement error. Ignoring measurement noise can substantially degrade statistical inference, while existing correction techniques are often computationally costly and inefficient. Recent advances in kernel methods, particularly those based on Maximum Mean Discrepancy (MMD), have enabled flexible, distribution-free inference, yet typically assume precise data and overlook contamination by measurement error. In this work, we introduce a novel framework for inference with samples corrupted by potentially heteroscedastic noise from a known distribution. Central to our approach is the convolutional MMD (convMMD), which compares distributions after noise convolution and retains metric validity under standard kernel conditions. We establish finite-sample deviation bounds that are unaffected by measurement error and prove an equivalence between testing under noise and kernel smoothing. Leveraging these insights, we introduce a convMMD-based estimator for inference with noisy, heteroscedastic observations. We establish its consistency and asymptotic normality, and provide an efficient implementation using stochastic gradient descent. We demonstrate the practical effectiveness of our approach through simulations and applications in astronomy and social sciences.