Beyond sparse denoising in frames: minimax estimation with a scattering transform

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 $\bf C^\alpha$ curves, wavelet, curvelet and Xlet frames are suboptimal if the Lipschitz exponent $\alpha \leq 2$ is an unknown parameter. Deep convolutional neural networks have recently obtained much better numerical results, which reach the minimax asymptotic bounds for all $\alpha$. 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 $\alpha \leq 2$. 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.