CIF: Medium: Collaborative Research: Estimating simultaneously structured models: from phase retrieval to network coding
National Science Foundation
08/15/2014 - 07/31/2018
This proposal focuses on a class of problems where several different low-dimensional structures are known to be present in the desired model. Existing literature has focused on estimating models with a single type of structure; in practice, however, there are many cases where the model is known to be structured in several ways. Applications include: (1) the sparse phase retrieval problem, a classical problem in signal processing and optical imaging, where the goal is to recover a sparse signal from "phaseless" measurements, (2) sparse PCA, a central problem in statistics and machine learning, where one seek approximate but sparse top eigenvectors so that the principal components are more interpretable than the usual PCA, and (3) code design for network communications, especially network coding. We propose a new mathematical framework to study simultaneously structured models", aiming to develop a united approach that explores simultaneous structures in all these problems and applications. At the same time, we draw on domain-specific knowledge to interpret our analysis, and to bridge between the different areas the problems come from. For example, we expect that we will build a tighter connection between coding theory and compressed sensing around low-rank matrix recovery problems. There is a need for new computationally tractable approaches for estimating these models.