The rapid development and availability of cheap storage and sensing devices has quickly produced a deluge of high-dimensional data. While the dimensionality of modern datasets continues to grow, our saving grace is that these data often exhibit low-dimensional structure that can be exploited to compress, organize, and cluster massive collections of data.

Signal models such as linear subspace models, remain one of the most widely used models for high-dimensional data; however, in many settings of interest, finding a global model that can capture all the relevant structure in the data is not possible. Thus, an alternative to learning a global model is to instead learn a hybrid model or a union of low-dimensional subspaces that model different subsets of signals in the dataset as living on distinct subspaces.

This thesis develops new methods and theory for learning union of subspace models as well as exploiting multi-subspace structure in a wide range of signal processing and data analysis tasks. The main contributions of this thesis include new methods and theory for: (i) decomposing and subsampling datasets consisting of signals on unions of subspaces, (ii) subspace clustering for learning union of subspace models, and (iii) exploiting multi-subspace structure in order accelerate distributed computing and signal processing on massive collections of data. I demonstrate the utility of the proposed methods in a number of important imaging and computer vision applications including: illumination-invariant face recognition, segmentation of hyperspectral remote sensing data, and compression of video and lightfield data arising in 3D scene modeling and analysis.