This award will support research on mathematical methods for signal representation and signal reconstruction in high dimensional problems. The project investigates randomized row-action methods as a tool for high dimensional signal recovery problems. Row-action algorithms are often well suited for large problems since they involve low complexity iterations that scale well in high dimensions. Moreover, the online nature of row-action methods is suitable for streaming data applications where one sequentially obtains access to individual parts of the entire system. Randomization is an essential tool for enabling row-action methods to yield provably fast and accurate results. The project will investigate randomized row-action methods for high dimensional applications such as signal reconstruction from frame coefficients, reconstruction from quantized samples, and sparse approximation. The project also studies the complementary problem of how to digitally encode information when high dimensionality places an enormous burden on physical devices, computational resources, and data storage. The project analyzes and designs efficient and robust analog-to-digital conversion algorithms for finite frames, compressed sensing problems, and ultra wideband interleaved sampling in non-synchronized environments.
The enormous size of modern data sets poses fundamental challenges to the ways in which data is analyzed, digitized, processed, and represented. Disparate problems such as computerized tomography, hyperspectral imaging, radar, and phase retrieval can involve signal classes that are so high dimensional that classically designed methods become impractical. This award will support the development of mathematical techniques to provide digital signal representations and signal reconstruction algorithms for emerging classes of high dimensional problems. The challenges of high dimensional data require approaches that go beyond linear signal representations and which are able to efficiently take advantage of nonlinearly structured signal classes and capitalize on very modest oversampling. A typical scenario occurs for high dimensional data such as streaming HD video that have information content of intrinsically lower dimension such as relatively few moving shapes. The analysis and algorithms in this project will apply broadly to modern signal processing techniques such as ultra wideband communications, compressed sensing problems, consistent reconstruction, analog-to-digital conversion and sigma-delta quantization. The award will support graduate student training, and graduate students will be involved in the project.
