This award supports the research program of the Principal Investigator in the general area of signal processing, which concerns itself, for example, with the electronic transmission of information: digitization, compression, encoding, decoding, and the like. Modern sensors are ubiquitous. They measure signals of interest and digitally store or transmit large amounts of data. Sophisticated computer algorithms then process the data, to recover the signals or to perform other tasks. To allow accurateÂ signal recovery and optimize subsequent use, it is critical that the measurement and digitization processes be well designed. It is also important thatÂ their theoretical properties, performance guarantees, and limitations be understood.Â Â Under practical constraints, weÂ theoretically investigate not only how measurements should be made, but also how they should be digitized (i.e., how they should be converted into bit-streams and subsequently compressed). The Principal Investigator will focus onÂ nonclassical signal models (including sparse vectors and low-rank matrices) whoseÂ importanceÂ has been increasing in recent years due to their relevance to modern applications.
This project aims to understand the interplay between sampling of structured signals, digitization, and compression---all in the modern setting of signal reconstruction by non-linear algorithms. Â For example, it focuses on approximately sparse signals measured using compressed sensing techniques, low-rank matrices measured by randomly sampling their entries, and vectors whose phase-less measurements consist of the magnitudes of inner products with other vectors. Â In short, the Principal Investigator seeks sampling and quantization theorems for the digital world. This requires developing and using tools from various areas of mathematics. As the measurement models related to this research rely heavily on randomness, the project must develop and use methods from geometric functional analysis and nonasymptotic random matrix theory. In the formulation of quantization methods for redundant measurements, connections with frame theory and the mathematical literature on noise-shaping quantization will be established and used. The encoding (compression) algorithms proposed in this research use randomness extensively, and here the Principal Investigator anticipates connections with the randomized numerical linear algebra literature. In analyzing new reconstruction algorithms, he will employ and develop methods from convex optimization and numerical analysis.