Title: Hunting geometric features in the probability density function with direct density-derivative-ratio estimation
Authors: Hiroaki Sasaki - Nara Institute of Science and Technology (Japan) [presenting]
Abstract: Geometric features in the probability density function underlying data is useful in statistical data analysis. For instance, the modes (i.e., local maxima) can be used for clustering, and the ridges unveil manifold structures hidden in data. A technical challenge to capture these geometric features is to estimate the ratio of the density derivatives to its density. A native approach to estimate the ``density-derivative-ratios'' is to first estimate the data density, then compute the derivatives of the estimated density, and finally take their ratios. However, this approach can be unreliable because a good density estimator does not necessarily mean a good density-derivative estimator. In addition, the division by the estimated density could magnify the estimation errors. To cope with this problem, a new estimator, which directly approximates the density-derivative-ratios without going through density estimation, is proposed. Then, with the developed estimator, we propose novel methods for mode-seeking clustering and density ridge estimation. The proposed methods are theoretically analysed. Finally, we numerically demonstrate that the proposed methods outperform existing methods especially for high-dimensional data.