Neural Empirical Bayes

We formulate a novel framework that unifies kernel density estimation and empirical Bayes, where we address a broad set of problems in unsupervised learning with a geometric interpretation rooted in the concentration of measure phenomenon. We start by energy estimation based on a denoising objective which recovers the original/clean data X from its measured/noisy version Y with empirical Bayes least squares estimator. The setup is rooted in kernel density estimation, but the log-pdf in Y is parametrized with a neural network, and crucially, the learning objective is derived for any level of noise/kernel bandwidth. Learning is efficient with double backpropagation and stochastic gradient descent. An elegant physical picture emerges of an interacting system of high-dimensional spheres around each data point, together with a globally-defined probability flow field. The picture is powerful: it leads to a novel sampling algorithm, a new notion of associative memory, and it is instrumental in designing experiments. We start with extreme denoising experiments. Walk-jump sampling is defined by Langevin MCMC walks in Y, along with asynchronous empirical Bayes jumps to X. Robbins associative memory is defined by a deterministic flow to attractors of the learned probability flow field. Finally, we observed the emergence of remarkably rich creative modes in the regime of highly overlapping spheres.