Nonlinear signal models: geometry, algorithms, and analysis

Traditional signal processing systems, based on linear modeling principles, face a stifling pressure to meet present-day demands caused by the deluge of data generated, transmitted and processed across the globe. Fortunately, recent advances have resulted in the emergence of more sophisticated, nonlinear signal models. Such nonlinear models have inspired fundamental changes in which information processing systems are designed and analyzed. For example, the sparse signal model serves as the basis for Compressive Sensing (CS), an exciting new framework for signal acquisition. In this thesis, we advocate a geometry-based approach for nonlinear modeling of signal ensembles. We make the guiding assumption that the signal class of interest forms a nonlinear low-dimensional manifold belonging to the high-dimensional signal space. A host of traditional nonlinear data models can be essentially interpreted as specific instances of such manifolds. Therefore, our proposed geometric approach provides a common framework that can unify, analyze, and significantly extend the scope of nonlinear models for information acquisition and processing. We demonstrate that the geometric approach enables new algorithms and analysis for a number of signal processing applications. Our specific contributions include: (i) new convex formulations and algorithms for the design of linear systems for data acquisition, compression, and classification; (ii) a general algorithm for reconstruction, deconvolution, and denoising of signals, images, and matrix-valued data; (iii) efficient methods for inference from a small number of linear signal samples, without ever resorting to reconstruction; and, (iv) new signal and image representations for robust modeling and processing of large-scale data ensembles.