Distortion Corrected Kernel Density Estimator on Riemannian Manifolds
针对流形学习嵌入扭曲几何的问题，提出一种畸变校正核密度估计器，利用黎曼度量信息提高密度估计精度，可用于异常检测，并在高维数据和爱尔兰智能电表数据上验证了效果。
Manifold learning obtains a low-dimensional representation of an underlying Riemannian manifold supporting high-dimensional data. Kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the Riemannian manifold. We propose a novel distortion-corrected kernel density estimator (DC-KDE) for any manifold learning embedding, with a bandwidth that depends on the estimated Riemannian metric at each data point. Exploiting the geometric information of the manifold leads to more accurate density estimation, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations including one with data lying in a 100 dimensional ambient space. We demonstrate that the proposed DC-KDE improves the density estimates as long as the manifold learning embedding is of sufficient quality, and has higher rank correlations with the true manifold density. Further simulation results are provided via a supplementary R shiny app. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data.