基于似然的深度学习在散斑回归中的极小极大理论
Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression
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中文总结 AI 辅助
研究乘性散斑噪声和加性高斯噪声模型下平滑非参数回归函数的极小极大估计，用基于似然的深度神经网络估计器，建立估计误差上界与极小极大下界，证明估计难度不变，数值实验验证方法有效性。
中文摘要 AI 辅助
散斑噪声是合成孔径雷达、光学相干断层扫描和数字全息等相干成像模态中常见的乘性噪声。深度学习方法在散斑去噪中取得了先进性能，但其统计极限未被充分探索。乘性散斑噪声使回归函数无法从条件均值识别，传统基于最小二乘法的深度学习方法不适用。本文研究在乘性散斑噪声和加性高斯噪声模型下，使用基于似然的深度神经网络估计器对平滑非参数回归函数的极小极大估计。建立了所提深度神经网络估计器估计误差的有限样本上界，并推导了非参数函数恢复的极小极大下界，表明它们在样本大小上相差对数因子。数值实验支持了基于深度神经网络的去散斑方法的一致性并证明了其有效性。
英文摘要
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.
