Fast Solution of $\ell _{1}$ -Norm Minimization Problems When the Solution May Be Sparse

The minimum lscr
₁
-norm solution to an underdetermined system of linear equations
y
=
Ax
is often, remarkably, also the sparsest solution to that system. This sparsity-seeking property is of interest in signal processing and information transmission. However, general-purpose optimizers are much too slow for lscr
₁
minimization in many large-scale applications.In this paper, the Homotopy method, originally proposed by Osborne et al. and Efron et al., is applied to the underdetermined lscr
₁
-minimization problem min par
x
par
₁
subject to
y
=
Ax
. Homotopy is shown to run much more rapidly than general-purpose LP solvers when sufficient sparsity is present. Indeed, the method often has the following
k
-step solution property: if the underlying solution has only
k
nonzeros, the Homotopy method reaches that solution in only
k
iterative steps. This
k
-step solution property is demonstrated for several ensembles of matrices, including incoherent matrices, uniform spherical matrices, and partial orthogonal matrices. These results imply that Homotopy may be used to rapidly decode error-correcting codes in a stylized communication system with a computational budget constraint. The approach also sheds light on the evident parallelism in results on lscr
₁
minimization and orthogonal matching pursuit (OMP), and aids in explaining the inherent relations between Homotopy, least angle regression (LARS), OMP, and polytope faces pursuit.