Compressed Representations of Signals and Models
In modern digital systems, the efficient representation of data becomes an important consideration. The encoding and decoding process of digital signals can be computationally expensive and can often induce significant errors. This dissertation studies three new methods in efficient data compression and recovery. The first, Stochastic Markov Gradient Descent, is a technique for training neural networks with small memory footprints. The second, Dynamical Quantization, exploits rigid structure available in frame theory to produce exponentially accurate quantized redundant frame expansions. The third and final method introduced in this dissertation is Look Ahead Thresholding which employs gradient-informed projections to enhance standard compressed sensing algorithms. Each of these methods is supported by rigorous mathematical theory explaining their function as well as practical experiments showcasing applicability to real problems.