An algorithm for data-driven bandwidth selection

The analysis of a feature space that exhibits multiscale patterns often requires kernel estimation techniques with locally adaptive bandwidths, such as the variable-bandwidth mean shift. Proper selection of the kernel bandwidth is, however, a critical step for superior space analysis and partitioning. This paper presents a mean shift-based approach for local bandwidth selection in the multimodal, multivariate case. Our method is based on a fundamental property of normal distributions regarding the bias of the normalized density gradient. We demonstrate that, within the large sample approximation, the local covariance is estimated by the matrix that maximizes the magnitude of the normalized mean shift vector. Using this property, we develop a reliable algorithm which takes into account the stability of local bandwidth estimates across scales. The validity of our theoretical results is proven in various space partitioning experiments involving the variable-bandwidth mean shift.