Minimum complexity pursuit

The fast growing field of compressed sensing is founded on the fact that if a signal is simple and has some 'structure', then it can be reconstructed accurately with far fewer samples than its ambient dimension. Many different plausible structures have been explored in this field, ranging from sparsity to low-rankness and to finite rate of innovation. However, there are important abstract questions that are yet to be answered. For instance, what are the general abstract meanings of structure and simplicity? Does there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension? In this paper, we aim to address these two questions. Using algorithmic informa- tion theory tools such as Kolmogorov complexity, we provide a unified method of describing simplicity and structure. We then explore the performance of an algorithm motivated by Ocams Razor (called MCP for minimum complexity pursuit) and show that it requires O(klogn) number of samples to recover a signal, where k and n represent its complexity and ambient dimension, respectively. Finally, we discuss more general classes of signals and provide guarantees on the performance of MCP.