Invariant multiscale statistics for inverse problems



We propose a new approach to linear ill-posed inverse problems. Our algorithm stabilizes the inversion by enforcing a new statistical constraint in a suitable feature space. We use the non-linear multiscale scattering transform—a complex convolutional network which discards the phase and thus exposes strong spectral correlations otherwise hidden beneath the phase fluctuations. We apply the algorithm to super-resolution and tomography with synthetic signals, and show that it outperforms regularized methods and stably recovers the missing spectrum. Further, we discuss the choice of the feature transform as a function of the operator and input statistics, and we prove convergence of the proposed iterative algorithm.


I Dokmanic
Joan Bruna
Stephane Mallat
M de Hoop