Abstract

We propose a new method for dimension reduction of high-dimensional spherical data based on the nonlinear projection of sphere-valued data to a randomly-chosen subsphere. The proposed method, spherical random projection, leads to a probabilistic lower-dimensional mapping of spherical data into a subsphere of the original space and is analogous the well-known concept of random projection on Euclidean space. In this paper, we investigate some properties of spherical random projection, including expectation preservation and distance concentration, from which we derive an analog of the Johnson-Lindenstrauss Lemma for spherical random projection. Clustering model selection is discussed as a statistical application of spherical random projection, and numerical experiments are conducted using both real and simulated data. Promising results from these experiments provide evidence for the usefulness of spherical random projection as a data analysis tool.