We present a numerical method to evaluate mutual information (MI) in nonlinear Gaussian noise channels by using denoising score matching (DSM) learning for estimating the score function of channel output. Via de Bruijn's identity, Fisher information estimated from the learned score function yields accurate estimates of MI through a Fisher integral representation for a variety of priors and channel nonlinearities. In this work, we propose a comprehensive theoretical foundation for the Score-to-Fisher bridge methodology, along with practical guidelines for its implementation. We also conduct extensive validation experiments, comparing our approach with closed-form solutions and a kernel density estimation baseline. The results of our numerical experiments demonstrate that the proposed method is both practical and efficient for MI estimation in nonlinear Gaussian noise channels. Additionally, we discuss the theoretical connections between our score-based framework and thermodynamic concepts, such as partition function estimation and optimal transport.

翻译：本文提出了一种数值方法，用于评估非线性高斯噪声信道中的互信息（MI）。该方法通过去噪评分匹配学习来估计信道输出的评分函数。借助德布鲁因恒等式，从学习到的评分函数估计出的费希尔信息，可通过费希尔积分表示为多种先验分布和信道非线性特性提供准确的互信息估计。本研究为评分-费希尔桥方法建立了完整的理论基础，并提供了实际实施指南。我们还进行了广泛的验证实验，将所提方法与闭式解及核密度估计基准进行比较。数值实验结果表明，该方法对于非线性高斯噪声信道中的互信息估计既实用又高效。此外，我们探讨了基于评分的框架与热力学概念（如配分函数估计和最优传输）之间的理论联系。