Efficient Manifold Approximation with Spherelets
提出用球面（spherelets）替代线性近似来逼近高维数据中的低维流形，开发了球面主成分分析（SPCA），理论证明其收敛速度更快、覆盖数更小，且能输出拟合值用于模型评估。
Abstract 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 mean squared errors 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.