{"ID":2851742,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.19612","arxiv_id":"2510.19612","title":"Beyond sparse denoising in frames: minimax estimation with a scattering transform","abstract":"A considerable amount of research in harmonic analysis has been devoted to non-linear estimators of signals contaminated by additive Gaussian noise. They are implemented by thresholding coefficients in a frame, which provide a sparse signal representation, or by minimising their $\\ell^1$ norm. However, sparse estimators in frames are not sufficiently rich to adapt to complex signal regularities. For cartoon images whose edges are piecewise $\\bf C^α$ curves, wavelet, curvelet and Xlet frames are suboptimal if the Lipschitz exponent $α\\leq 2$ is an unknown parameter. Deep convolutional neural networks have recently obtained much better numerical results, which reach the minimax asymptotic bounds for all $α$. Wavelet scattering coefficients have been introduced as simplified convolutional neural network models. They are computed by transforming the modulus of wavelet coefficients with a second wavelet transform. We introduce a denoising estimator by jointly minimising and maximising the $\\ell^1$ norms of different subsets of scattering coefficients. We prove that these $\\ell^1$ norms capture different types of geometric image regularity. Numerical experiments show that this denoising estimator reaches the minimax asymptotic bound for cartoon images for all Lipschitz exponents $α\\leq 2$. We state this numerical result as a mathematical conjecture. It provides a different harmonic analysis approach to suppress noise from signals, and to specify the geometric regularity of functions. It also opens a mathematical bridge between harmonic analysis and denoising estimators with deep convolutional network.","short_abstract":"A considerable amount of research in harmonic analysis has been devoted to non-linear estimators of signals contaminated by additive Gaussian noise. They are implemented by thresholding coefficients in a frame, which provide a sparse signal representation, or by minimising their $\\ell^1$ norm. However, sparse estimator...","url_abs":"https://arxiv.org/abs/2510.19612","url_pdf":"https://arxiv.org/pdf/2510.19612v2","authors":"[\"Nathanaël Cuvelle--Magar\",\"Stéphane Mallat\"]","published":"2025-10-22T14:05:25Z","proceeding":"cs.CV","tasks":"[\"cs.CV\"]","methods":"[]","has_code":false}