This paper studies how well deep neural networks can recover signals that are corrupted by speckle. Speckle is a multiplicative noise that appears in coherent imaging systems such as synthetic aperture radar, optical coherence tomography and digital holography. The authors point out a key difficulty: because speckle multiplies the signal, the conditional mean of the observations is zero and the usual least-squares training used in many denoising methods no longer identifies the underlying signal.

To address this, the researchers propose training deep neural networks by directly minimizing a likelihood-based loss derived from a simple statistical model. The model for each observation is yi = f*(xi) ξi + τi, where f*(x) is the unknown signal function to recover, ξi models speckle noise (treated here as Gaussian by a standard approximation) and τi is additive Gaussian measurement noise. The negative log-likelihood used for training is an average of terms of the form y_i^2/(f(x_i)^2+σ_τ^2) + log(f(x_i)^2+σ_τ^2). For high-dimensional inputs that depend only on a few relevant coordinates, they also propose an ℓ1-penalized version that combines variable selection with function estimation.

The main theoretical results are finite-sample upper bounds on the estimation error of these neural-network estimators and matching minimax lower bounds that show those upper bounds are near-optimal up to logarithmic factors in the sample size. In particular, for a class of smooth target functions the mean squared error scales like n^{-2γ*/(2γ*+1)} (up to logs), where γ* encodes an adjusted smoothness of the target. The minimax lower bound shows no estimator can do substantially better in the worst case, so the proposed method is close to the best possible under the model.

Why this matters: the results show that, despite the extra complication of multiplicative speckle, estimating a smooth signal is not intrinsically harder than the well-studied case of additive Gaussian noise. In other words, carefully designed likelihood-based deep learning can achieve essentially the same statistical rates as for additive noise. That gives a theoretical foundation for using deep learning in practical despeckling tasks in imaging modalities that suffer from speckle.