Didong Li : Subspace Approximations with Spherelets

Data lying in a high-dimensional ambient space are commonly thought to have a much lower intrinsic dimension. In particular, the data may be concentrated near a lower-dimensional subspace or manifold. There is an immense literature focused on approximating the unknown subspace, and in exploiting such approximations in clustering, data compression, and building of predictive models. Most of the literature relies on approximating subspaces using a locally linear, and potentially multiscale, dictionary. In this talk, we propose a simple and general alternative, which instead uses pieces of spheres, or spherelets, to locally approximate the unknown subspace. Theory is developed showing that spherelets can produce dramatically lower covering numbers and MSEs for many manifolds. We develop spherical principal components analysis (SPCA) and spherical multiscale methods. Results relative to state-of-the-art competitors show dramatic gains in ability to accurately approximate the subspace with orders of magnitude fewer components. This leads to substantial gains in data compressibility, few clusters and hence better interpretability, and much lower MSE based on small to moderate sample sizes. A Bayesian nonparametric model based on spherelets will be introduced as an application.

• Category: Graduate/Faculty Seminar
• Duration: 01:14:58
• Date:  April 2, 2018 at 11:55 AM
• Views: 150
• Tags:   seminar, Graduate/faculty Seminar