Denoising data using convex relaxations
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914533269831680 |
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| author | Fefferman, Charles Gangopadhyay, Aalok Lassas, Matti Marty, Jonathan Narayanan, Hariharan |
| author_facet | Fefferman, Charles Gangopadhyay, Aalok Lassas, Matti Marty, Jonathan Narayanan, Hariharan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02327 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Denoising data using convex relaxations Fefferman, Charles Gangopadhyay, Aalok Lassas, Matti Marty, Jonathan Narayanan, Hariharan Methodology Machine Learning 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. |
| title | Denoising data using convex relaxations |
| topic | Methodology Machine Learning |
| url | https://arxiv.org/abs/2605.02327 |