Structured low-rank matrix recovery via optimization methods

Abstract

From single-molecule microscopy in biology, to collaborative filtering in recommendation systems, to quantum state tomography in physics, many scientific discoveries involve solving ill-posed inverse problems, where the number of parameters to be estimated far exceeds the number of available measurements. To make these daunting problems solvable, low-dimensional geometric structures are often exploited, and regularizations that promote underlying structures are used for various inference tasks. To date, one of the most effective and plausible low-dimensional models for matrix data is the low-rank structure, which assumes that columns of the data matrix are correlated and lie in a low-dimensional subspace. This helps make certain matrix inverse problems well-posed. However, in some cases, standard low-rank structure is not powerful enough for modeling the underlying data generating process, and additional modeling efforts are desired. This is the main focus of this research. Motivated by applications from different disciplines in engineering and science, in this dissertation, we consider the recovery of three instances of structured matrices from limited measurement data, where additional structures naturally occur in the data matrices beyond simple low-rankness. The structured matrices that we consider include i) low-rank and spectrally sparse matrices in super-resolution imaging; ii) low-rank skew-symmetric matrices in pairwise comparisons; iii) and low-rank positive semidefinite matrices in physical and data sciences. Using optimization as a tool, we develop new regularizers and computationally efficient algorithmic frameworks to account for structured low-rankness in solving these ill-posed inverse problems. For some of the problems considered in this dissertation, theoretical analysis is also carried out for the proposed optimization programs. We show that, under mild conditions, the structured low-rank matrices can be recovered reliably from a minimal number of random measurements.