Nonparametric Deconvolution and Denoising using Simulation Based Inference

Abstract

We address the challenge of recovering latent signals obscured by measurement noise. We propose a likelihood-free framework for nonparametric density deconvolution and empirical Bayes denoising under additive measurement error. Our approach uses convolutional maximum mean discrepancy loss to match observed data distributions to noise-convolved model distributions, enabling compatibility with expressive generative models like Gaussian mixtures and normalizing flows. Theoretically, this work extends convMMD from parametric to nonparametric estimation, establishing finite-sample bounds for empirical sieve minimizers and L₂ convergence rates under Sobolev smoothness. The convergence rates reflect classical inverse-problem characteristics, demonstrating polynomial rates for ordinary-smooth noise and logarithmic rates for super-smooth noise. The learned density functions as an empirical prior for denoising individual latent values, offering a practical, theoretically grounded approach to deconvolution under generative latent distribution models.

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