537 research outputs found
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
A Proximal-Gradient Homotopy Method for the Sparse Least-Squares Problem
We consider solving the -regularized least-squares (-LS)
problem in the context of sparse recovery, for applications such as compressed
sensing. The standard proximal gradient method, also known as iterative
soft-thresholding when applied to this problem, has low computational cost per
iteration but a rather slow convergence rate. Nevertheless, when the solution
is sparse, it often exhibits fast linear convergence in the final stage. We
exploit the local linear convergence using a homotopy continuation strategy,
i.e., we solve the -LS problem for a sequence of decreasing values of
the regularization parameter, and use an approximate solution at the end of
each stage to warm start the next stage. Although similar strategies have been
studied in the literature, there have been no theoretical analysis of their
global iteration complexity. This paper shows that under suitable assumptions
for sparse recovery, the proposed homotopy strategy ensures that all iterates
along the homotopy solution path are sparse. Therefore the objective function
is effectively strongly convex along the solution path, and geometric
convergence at each stage can be established. As a result, the overall
iteration complexity of our method is for finding an
-optimal solution, which can be interpreted as global geometric rate
of convergence. We also present empirical results to support our theoretical
analysis
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