Efficient Manifold and Subspace Approximations with Spherelets
This provides a more efficient alternative for dimensionality reduction in statistical applications, though it appears incremental as it builds on existing manifold approximation literature.
The paper tackles the problem of approximating lower-dimensional manifolds in high-dimensional data by proposing spherelets, a method using spheres instead of linear approximations, and shows that it achieves lower covering numbers and MSEs with fewer components compared to state-of-the-art competitors.
In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold. There is a rich literature on approximating the unknown manifold, and on exploiting such approximations in clustering, data compression, and prediction. Most of the literature relies on linear or locally linear approximations. In this article, we propose a simple and general alternative, which instead uses spheres, an approach we refer to as spherelets. We develop spherical principal components analysis (SPCA), and provide theory on the convergence rate for global and local SPCA, while showing that spherelets can provide lower covering numbers and MSEs for many manifolds. Results relative to state-of-the-art competitors show gains in ability to accurately approximate manifolds with fewer components. Unlike most competitors, which simply output lower-dimensional features, our approach projects data onto the estimated manifold to produce fitted values that can be used for model assessment and cross validation. The methods are illustrated with applications to multiple data sets.