Convolutional Maximum Mean Discrepancy for Inference in Noisy Data

TL;DR

This paper introduces convMMD, a kernel-based method for statistical inference on noisy data that remains robust and efficient despite measurement errors, with theoretical guarantees and practical applications.

Contribution

It develops a novel convolutional MMD framework that corrects for measurement noise, providing consistent, asymptotically normal estimators with efficient implementation.

Findings

Finite-sample deviation bounds unaffected by measurement error

ConvMMD-based estimator is consistent and asymptotically normal

Effective in astronomy and social science data analysis

Abstract

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.