Dimension Reduction Algorithms for Near - Optimal Low - Dimensional Embeddings and Compressive Sensing

In this thesis, we establish theoretical guarantees for several dimension reduction algorithms developed for applications in compressive sensing and signal processing. In each instance, the input is a point or set of points in d-dimensional Euclidean space, and the goal is to find a linear function from Rd into Rk , where k << d, such that the resulting embedding of the input pointset into k-dimensional Euclidean space has various desirable properties. We focus on two classes of theoretical results: " First, we examine linear embeddings of arbitrary pointsets with the aim of minimizing distortion. We present an exhaustive-search-based algorithm that yields a kdimensional linear embedding with distortion at most eot(k) + 6 for any 6 > 0, where opt(k) is the smallest possible distortion over all orthonormal embeddings into k dimensions. This PTAS-like result transcends lower bounds for well-known embedding teclhniques such as the Johnson-Lindenstrauss transform. " Next, motivated by compressive sensing of images, we examine linear embeddings of datasets containing points that are sparse in the pixel basis, with the goal of recoving a nearly-optimal sparse approximation to the original data. We present several algorithms that achieve strong recovery guarantees using the near-optimal bound of O(k log n) measurements, while also being highly "local" so that they can be implemented more easily in physical devices. We also present some impossibility results concerning the existence of such embeddings with stronger locality properties. Thesis Supervisor: Piotr Indyk Title: Professor of Electrical Engineering and Computer Science