Adaptive manifold density estimation

Abstract

When data concentrate on lower dimensional manifolds embedded in a higher dimensional representation space and these manifolds have different intrinsic geometry and dimensions, density estimators with locally adaptive smoothing parameters show substantial gains over those with fixed bandwidths. However, it is difficult to decide how the local bandwidth parameters should be parameterized. In this paper, we propose a sample-point density estimator. For each observation, we associate it with a unique bandwidth parameter which is achieved by computing the local covariance matrix of its k nearest neighbours. Compared to the fixed choice of the neighbourhood size for all the samples in [Vincent P, Bengio Y. Manifold parzen windows. In: Becker S, Thrun S, and Obermayer K, editors, Advances in neural information processing systems Vol. 15. MIT Press; 2003. p. 849–856], our adaptive manifold density estimator allows samples to select their own neighbourhood sizes.