Model-free filtering in high dimensions via projection and score-based diffusions

Abstract

We consider the problem of recovering a latent signal $X$ from its noisy observation $Y$. The unknown law ${\mathbb{P}}^X$ of $X$, and in particular its support $\mathcal{M}$, are accessible only through a large sample of i.i.d.\ observations. We further assume $\mathcal{M}$ to be a low-dimensional submanifold of a high-dimensional Euclidean space ${\mathbb{R}}^d$. As a filter or denoiser $\hat{X}$, we suggest an estimator of the metric projection ${\pi }_{\mathcal{M}}(Y)$ of $Y$ onto the manifold $\mathcal{M}$. To compute this estimator, we study an auxiliary semiparametric model in which $Y$ is obtained by adding isotropic Laplace noise to $X$. Using score matching within a corresponding diffusion model, we obtain an estimator of the Bayesian posterior ${\mathbb{P}}^{X\mid Y}$ in this setup. Our main theoretical results show that, in the limit of high dimension $d$, this posterior ${\mathbb{P}}^{X\mid Y}$ is concentrated near the desired metric projection ${\pi }_{\mathcal{M}}(Y)$.