Dimensionality Reduction with Spherical Constraints

“Curse of dimensionality” has been a significant obstacle to solving many problems. One way to avoid this obstacle is to use dimensionality reduction methods to reduce the dimension of the data while preserving the properties of the data. Reader is referred to [Saul, 2005, Fodor, 2002] for a detailed review of these dimensionality reduction methods. Almost all of these dimensionality reduction methods embed the data into a lower dimensional subspace without assuming anything about the structure of the embedding space. There are problems, in which one wants the embedding space to have a special structure i.e. spherical structure. More specifically, if points lie on a sphere in the original space, points in the lower dimensional subspace should also lie on a sphere. In this work, we aim at solving exactly this problem. We reduce the dimensionality of the data that lie on a higher dimensional sphere while maintaining this spherical structure.