Mutual Information Estimation via Score-to-Fisher Bridge for Nonlinear Gaussian Noise Channels

We present a numerical method to evaluate mutual information (MI) in nonlinear Gaussian noise channels by using denoising score matching learning for estimating the score function of channel output. Via de Bruijn identity and the I--MMSE relation, Fisher information estimated from the learned score yields accurate estimates of MI through an integral representation of MI for a variety of priors and channel nonlinearities (e.g., elementwise nonlinearity). 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 mutual information estimation in nonlinear Gaussian noise channels.