Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression

TL;DR

The paper develops a likelihood-based deep learning framework for regression under multiplicative speckle noise combined with additive Gaussian noise, establishing tight finite-sample error bounds and minimax rates. It shows that the estimation difficulty remains comparable to standard additive noise settings, despite the challenges of multiplicative noise. The proposed DNN estimators are effective across low- and high-dimensional feature spaces.

Why It Matters

Speckle noise severely degrades image quality in medical and remote sensing imaging, and this work provides a statistically sound foundation for deep learning methods in such settings. It validates the use of likelihood-based DNNs for speckle regression, offering theoretical guarantees that support practical deployment in real-world applications.

Key Findings

• Likelihood-based DNN estimators achieve minimax optimal rates for nonparametric regression under mixed multiplicative and additive noise.
• The minimax rates match those for additive Gaussian noise alone, up to logarithmic factors, indicating no significant increase in estimation difficulty.
• The framework is applicable to both low-dimensional and sparse high-dimensional feature settings, with numerical results confirming practical effectiveness.