Learning-based Compressive Subsampling

The problem of recovering a structured signal x ∈ mathbbCp from a set of dimensionality-reduced linear measurements b = mathbf Amathbf x arises in a variety of applications, such as medical imaging, spectroscopy, Fourier optics, and computerized tomography. Due to computational and storage complexity or physical constraints imposed by the problem, the measurement matrix A ∈ mathbbCn × p is often of the form A = POmegaboldsymbolPsi for some orthonormal basis matrix boldsymbolPsi∈ mathbbCp × p and subsampling operator POmega: mathbbCp rightarrow mathbbCn that selects the rows indexed by Omega. This raises the fundamental question of how best to choose the index set Omega in order to optimize the recovery performance. Previous approaches to addressing this question rely on non-uniform random subsampling using application-specific knowledge of the structure of x. In this paper, we instead take a principled learning-based approach in which a fixed index set is chosen based on a set of training signals x1,dotsc,xm. We formulate combinatorial optimization problems seeking to maximize the energy captured in these signals in an average-case or worst-case sense, and we show that these can be efficiently solved either exactly or approximately via the identification of modularity and submodularity structures. We provide both deterministic and statistical theoretical guarantees showing how the resulting measurement matrices perform on signals differing from the training signals, and we provide numerical examples showing our approach to be effective on a variety of data sets.