This paper revisits the 70-year-old shrinkage problem and establishes a precise connection among three subjects: higher-order Fisher-type information, optimal transport, and the combinatorics of integer partitions.
Empirical Bayes tends to produce overly aggressive shrinkage as a denoiser. We introduce new denoisers that optimally shrink the distribution toward the true signal distribution with order-of-magnitude improvements. Unlike empirical Bayes denoiser, our denoisers are universal and agnostic to the signal and noise distributions. One immediate application of our distributional shrinkage theory is to enhance generative modeling: we can replace the stochastic backward diffusion process with optimal deterministic denoisers to achieve higher-order accuracy.