Geometric Models for Dimensionality Reduction in Signal and Data Processing

The ever-growing bulk and dimensionality of signals and data places an increasing burden on every stage of the data processing pipeline, from the initial data acquisition to the subsequent transmission, storage, and analysis. Fortunately, this burden can often be alleviated (the "dimensionality" effectively reduced) by exploiting concise models for signal structure. For example, many signals have a sparse representation in terms of some dictionary, while certain signal families tend to cluster along low-dimensional manifolds within the ambient signal space. In this talk I will overview several recent advances in dimensionality reduction, including new insight and analysis into the geometry of low-dimensional signal models and radically new methods for dimensionality reduction inspired by these models. I will pay particular attention to geometric principles that underly the emerging theory of Compressive Sensing (CS), which states that a sparse signal can be recovered from a small number of random linear measurements. Building on these ideas, I will explain how and why CS can be extended beyond the framework of sparsity to include more general manifold-based models. I will also discuss the surprising lack of differentiability that arises in certain image manifolds corresponding to seemingly innocuous image families, the consequences of this nondifferentiability in solving fundamental problems in image processing, and a multiscale tangent characterization that allows very efficient navigation along such manifolds.
Michael B. Wakin received the B.S. degree in electrical engineering and the B.A. degree in mathematics in 2000 (summa cum laude), the M.S. degree in electrical engineering in 2002, and the Ph.D. degree in electrical engineering in 2007, all from Rice University. He is currently an NSF Mathematical Sciences Postdoctoral Research Fellow in the Department of Applied and Computational Mathematics at the California Institute of Technology. His research interests include sparse, geometric, and manifold-based models for signal and image processing, approximation, compression, compressive sensing, and dimensionality reduction.