Multivariate locally adaptive kernel density estimation

When the underlying density exhibits multiple modes with different scales and orientations, density estimators with locally adaptive smoothing parameters show substantial gains over those with fixed bandwidths. However, it is a concern that the local smoothing matrices may not be well parameterized, and the corresponding optimization problems will be difficult. In this paper, we build a more promising and practical algorithm. The local bandwidth factors are chosen through clustering, and the global smoothing parameter is achieved by optimizing the Asymptotic Mean Integrated Squared Error. Most importantly, our locally adaptive estimator involves optimizing a scalar rather than solving a costly multivariate optimization problem. Our method, which can also be applied to manifold density estimation, is an improvement and generalization of the binned version estimator of Sain.