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 ℓ1 norm. However, sparse estimators in frames are not sufficiently rich to adapt to complex signal regularities. For cartoon images whose edges are piecewise bf Cα curves, wavelet, curvelet and Xlet frames are suboptimal if the Lipschitz exponent α ≤ 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 ℓ1 norms of different subsets of scattering coefficients. We prove that these ℓ1 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 ≤ 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.