Denoising data using convex relaxations

TL;DR

This paper introduces a convex relaxation method for denoising data sampled from a low-dimensional manifold in high-dimensional space, combining PCA and convex hull projection with theoretical guarantees.

Contribution

It proposes a novel convex-relaxation estimator that leverages empirical Gaussian tail probabilities to improve denoising of manifold-structured data.

Findings

Finite-sample guarantees for the denoising estimator.

Error bounds derived under a lower-mass condition.

Application to Cryo-EM data with verified assumptions.

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.