# Estimating Local Function Complexity via Mixture of Gaussian Processes.

Abstract

Real world data often exhibit inhomogeneity, e.g., skewed distribution, non-uniform complexity of the target function and uneven noise level over the input space. In this paper, we cope with inhomogeneity by explicitly estimating the locally optimal kernel bandwidth as a function. Specifically, we propose Spatially Adaptive Bandwidth Estimation in Regression (SABER), which employs the mixture of experts consisting of multinomial kernel logistic regression as a gate and Gaussian process regression models as experts. SABER can estimate the optimal kernel bandwidth much more accurately and stably than existing kernel width estimation methods, and shows comparable prediction performance with deep neural networks in quantum chemistry applications. Drawing parallels to the theory of locally linear smoothing, we derive an estimate to the local function complexity. Local function complexity can be used for model interpretation, active learning and Bayesian optimization. Those aspects are also demonstrated in quantum chemistry experiments and fluid dynamics simulations.