Beyond sparse denoising in frames: minimax estimation with a scattering transform
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912669643046912 |
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| author | Cuvelle--Magar, Nathanaël Mallat, Stéphane |
| author_facet | Cuvelle--Magar, Nathanaël Mallat, Stéphane |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond sparse denoising in frames: minimax estimation with a scattering transform Cuvelle--Magar, Nathanaël Mallat, Stéphane Computer Vision and Pattern Recognition 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. |
| title | Beyond sparse denoising in frames: minimax estimation with a scattering transform |
| topic | Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2510.19612 |