Learning-based Compressive Subsampling

• Main Authors: Baldassarre, Luca, Li, Yen-Huan, Scarlett, Jonathan, Gözcü, Baran, Bogunovic, Ilija, Cevher, Volkan
• Format: Journal Article
• Language: English
• Published: 28-03-2016
• Subjects: Computer Science - Information Theory; Computer Science - Learning; Mathematics - Information Theory; Statistics - Machine Learning
• DOI: 10.48550/arxiv.1510.06188

The problem of recovering a structured signal $\mathbf{x} \in \mathbb{C}^p$
from a set of dimensionality-reduced linear measurements $\mathbf{b} = \mathbf
{A}\mathbf {x}$ 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 $\mathbf{A} \in \mathbb{C}^{n \times p}$ is often of the
form $\mathbf{A} = \mathbf{P}_{\Omega}\boldsymbol{\Psi}$ for some orthonormal
basis matrix $\boldsymbol{\Psi}\in \mathbb{C}^{p \times p}$ and subsampling
operator $\mathbf{P}_{\Omega}: \mathbb{C}^{p} \rightarrow \mathbb{C}^{n}$ 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 \emph{random} subsampling using application-specific knowledge of
the structure of $\mathbf{x}$. In this paper, we instead take a principled
learning-based approach in which a \emph{fixed} index set is chosen based on a
set of training signals $\mathbf{x}_1,\dotsc,\mathbf{x}_m$. 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.