Denoising data using convex relaxations

2026-05-04

AI Summary

This study addresses the problem of recovering latent signals lying on a low-dimensional manifold from high-dimensional observations corrupted by isotropic Gaussian noise. The authors propose a convex relaxation–based denoising method that first reduces dimensionality via principal component analysis and then projects the data onto the convex hull of the underlying manifold. A statistical oracle, constructed using empirical Gaussian tail probabilities, defines supporting hyperplanes to guide the estimation. This approach uniquely integrates convex hull projection with manifold structure for denoising, establishing finite-sample risk and error bounds by combining tools from convex geometry and statistical learning theory. The theoretical analysis leverages entropy of convex sets, covering numbers, and Lipschitz properties, and proves finite-sample consistency of the oracle under mild distributional assumptions. Experimental results in cryo-electron microscopy imaging validate the method’s efficacy and support the theoretical claims.

Abstract

We study the problem of denoising observations \(Y_i=X_i+Z_i\), where the latent variables \(X_i\) are sampled from a low-dimensional manifold in \(\mathbb{R}^n\) and the noise variables \(Z_i\) are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

Problem

• denoising
• low-dimensional manifold
• isotropic Gaussian noise
• convex relaxation
• latent variables

Innovation

• convex relaxation
• manifold denoising
• convex hull projection
• finite-sample guarantees
• Cryo-EM