Summary: | The past decade has witnessed the emergence of compressed sensing as
a way of acquiring sparsely representable signals in a compressed
form. These developments have greatly motivated research in sparse
signal recovery, which lies at the heart of compressed sensing, and
which has recently found its use in altogether new applications.
In the first part of this thesis we study the theoretical aspects of
joint-sparse recovery by means of sum-of-norms minimization, and the
ReMBo-l₁ algorithm, which combines boosting techniques with
l₁-minimization. For the sum-of-norms approach we derive
necessary and sufficient conditions for recovery, by extending
existing results to the joint-sparse setting. We focus in particular
on minimization of the sum of l₁, and l₂ norms, and give
concrete examples where recovery succeeds with one formulation but
not with the other. We base our analysis of ReMBo-l₁ on its
geometrical interpretation, which leads to a study of orthant
intersections with randomly oriented subspaces. This work
establishes a clear picture of the mechanics behind the method, and
explains the different aspects of its performance.
The second part and main contribution of this thesis is the
development of a framework for solving a wide class of convex
optimization problems for sparse recovery. We provide a detailed
account of the application of the framework on several problems, but
also consider its limitations. The framework has been implemented in
the SPGL1 algorithm, which is already well established as an
effective solver. Numerical results show that our algorithm is
state-of-the-art, and compares favorably even with solvers for the
easier---but less natural---Lagrangian formulations.
The last part of this thesis discusses two supporting software
packages: Sparco, which provides a suite of test problems for
sparse recovery, and Spot, a Matlab toolbox for the creation and
manipulation of linear operators. Spot greatly facilitates rapid
prototyping in sparse recovery and compressed sensing, where linear
operators form the elementary building blocks.
Following the practice of reproducible research, all code used for
the experiments and generation of figures is available online at
http://www.cs.ubc.ca/labs/scl/thesis/09vandenBerg/. === Science, Faculty of === Computer Science, Department of === Graduate
|