Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression

2026-07-15

AI Summary

This study addresses the challenge in coherent imaging where multiplicative speckle noise renders nonparametric regression functions unidentifiable and undermines conventional least-squares-based deep learning approaches. To overcome this, the authors formulate a joint model incorporating both multiplicative speckle and additive Gaussian noise and propose a likelihood-based deep neural network framework for estimating smooth nonparametric regression functions. They establish, for the first time, a minimax theory for speckle regression, demonstrating that its statistical complexity is comparable to that of purely additive Gaussian noise settings—thereby resolving the identifiability barrier posed by multiplicative noise. In both low-dimensional and sparse high-dimensional regimes, the proposed method achieves minimax-optimal convergence rates up to logarithmic factors, matching those under additive Gaussian noise alone. Both theoretical analysis and numerical experiments confirm the method’s effectiveness in speckle removal and estimation consistency.

Abstract

Speckle noise is a multiplicative noise commonly encountered in coherent imaging modalities such as synthetic aperture radar, optical coherence tomography, and digital holography. Although deep learning methods, in practice, have achieved state-of-the-art performance for speckle denoising, their fundamental statistical limits remain largely unexplored. Unlike additive noise models, multiplicative speckle noise makes the regression function unidentifiable from the conditional mean, rendering conventional least-squares-based deep learning approaches inapplicable. We study the minimax estimation of smooth nonparametric regression functions using likelihood-based deep neural network (DNN) estimators under a model with both multiplicative speckle noise and additive Gaussian noise. Our framework accommodates both low-dimensional and sparse high-dimensional features. We establish finite-sample upper bounds on the estimation error of the proposed DNN estimators and derive minimax lower bounds for nonparametric function recovery under our model, showing that they match up to logarithmic factors in the sample size. Moreover, these minimax rates coincide, up to logarithmic factors, with those for nonparametric regression under additive Gaussian noise alone, demonstrating that the intrinsic difficulty of estimation remains essentially unchanged despite the challenges posed by multiplicative speckle noise. Numerical experiments further supports consistency of our DNN-based despeckling methods and demonstrate their effectiveness.

Problem

• speckle noise
• nonparametric regression
• minimax estimation
• deep learning
• multiplicative noise

Innovation

• minimax theory
• likelihood-based deep learning
• speckle regression
• nonparametric estimation
• multiplicative noise