Sparse Modeling for High - Dimensional Multi - Manifold Data Analysis

High-dimensional data are ubiquitous in many areas of science and engineering, such as machine learning, signal and image processing, computer vision, pattern recognition, bioinformatics, etc. Often, high-dimensional data are not distributed uniformly in the ambient space; instead they lie in or close to a union of low-dimensional manifolds. Recovering such low-dimensional structures in the data helps to not only significantly reduce the computational cost and memory requirements of algorithms that deal with the data, but also reduce the effect of the high-dimensional noise in the data and improve the performance of inference and learning tasks. There are three fundamental tasks related to the multi-manifold data: clustering, dimensionality reduction, and classification. While the area of machine learning has seen great advances in these areas, the applicability of current algorithms are limited due to several challenges. First, in many problems, manifolds are spatially close or even intersect, while existing methods work only when manifolds are sufficiently separated. Second, most algorithms require to know the dimensions or the number of manifolds a priori, while in real-world problems such quantities are often unknown. ii