Denoising data using convex relaxations

• URL: http://arxiv.org/abs/2605.02327v2
• Date: Wed, 06 May 2026 06:45:33 GMT
• Title: Denoising data using convex relaxations
• Abstract summary: 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 (mathbbRn) and the noise variables (Z_i) are isotropic Gaussian.<n>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.
• Score: 4.069088378764892
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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