Implicitly Normalized Explicitly Regularized Density Estimation

We propose a new approach to non-parametric density estimation, that is based on regularizing a Sobolev norm of the density. This method is provably different from Kernel Density Estimation, and makes the bias of the model clear and interpretable. While there is no closed analytic form for the associated kernel, we show that one can approximate it using sampling. The optimization problem needed to determine the density is non-convex, and standard gradient methods do not perform well. However, we show that with an appropriate initialization and using natural gradients, one can obtain well performing solutions. Finally, while the approach provides unnormalized densities, which prevents the use of log-likelihood for cross validation, we show that one can instead adapt Fisher Divergence based Score Matching methods for this task. We evaluate the resulting method on the comprehensive recent Anomaly Detection benchmark suite, ADBench, and find that it ranks second best, among more than 15 algorithms.