{"ID":2849569,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.23197","arxiv_id":"2510.23197","title":"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 $\\mathscr{M}$, are accessible only through a large sample of i.i.d.\\ observations. We further assume $\\mathscr{M}$ to be a low-dimensional submanifold of a high-dimensional Euclidean space $\\mathbb{R}^d$. As a filter or denoiser $\\widehat X$, we suggest an estimator of the metric projection $π_{\\mathscr{M}}(Y)$ of $Y$ onto the manifold $\\mathscr{M}$. 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 $π_{\\mathscr{M}}(Y)$.","short_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 $\\mathscr{M}$, are accessible only through a large sample of i.i.d.\\ observations. We further assume $\\mathscr{M}$ to be a low-dimensional submanifold of a high-...","url_abs":"https://arxiv.org/abs/2510.23197","url_pdf":"https://arxiv.org/pdf/2510.23197v1","authors":"[\"Sören Christensen\",\"Jan Kallsen\",\"Claudia Strauch\",\"Lukas Trottner\"]","published":"2025-10-27T10:34:46Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ML\"]","methods":"[\"Diffusion Model\"]","has_code":false}