Neural Empirical Bayes

We unify kernel density estimation and empirical Bayes and address a set of problems in unsupervised machine learning with a geometric interpretation of those methods, rooted in the concentration of measure phenomenon. Kernel density is viewed symbolically as X Y where the random variable X is smoothed to Y = X + N (0, σ 2 I d), and empirical Bayes is the machinery to denoise in a least-squares sense, which we express as X Y. A learning objective is derived by combining these two, symbolically captured by X Y. Crucially, instead of using the original nonparametric estimators, we parametrize the energy function with a neural network denoted by φ; at optimality, ∇φ ≈ −∇ log f where f is the density of Y. The optimization problem is abstracted as interactions of high-dimensional spheres which emerge due to the concentration of isotropic Gaussians. We introduce two algorithmic frameworks based on this machinery: (i) a "walk-jump" sampling scheme that combines Langevin MCMC (walks) and empirical Bayes (jumps), and (ii) a probabilistic framework for associative memory, called NEBULA, definedà la Hopfield by the gradient flow of the learned energy to a set of attractors. We finish the paper by reporting the emergence of very rich "creative memories" as attractors of NEBULA for highly-overlapping spheres.