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arXiv · 2605.02327

Denoising data using convex relaxations

Charles Fefferman·Aalok Gangopadhyay·Matti Lassas·Jonathan Marty·Hariharan Narayanan
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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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Charles Fefferman, Aalok Gangopadhyay, Matti Lassas, Jonathan Marty, Hariharan Narayanan. 2026-05-06. Denoising data using convex relaxations. https://arxiv.org/abs/2605.02327

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