Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
Foundations of Computational Mathematics
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
SIAM Journal on Optimization
Probing the Pareto Frontier for Basis Pursuit Solutions
SIAM Journal on Scientific Computing
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
SIAM Journal on Imaging Sciences
Smoothing Techniques for Computing Nash Equilibria of Sequential Games
Mathematics of Operations Research
SIAM Journal on Scientific Computing
NESTA: A Fast and Accurate First-Order Method for Sparse Recovery
SIAM Journal on Imaging Sciences
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
SIAM Journal on Scientific Computing
| Hi-index | 0.00 |
We propose a first-order smoothed penalty algorithm (SPA) to solve the sparse recovery problem $\min\{\|x\|_1:Ax=b\}$. SPA is efficient as long as the matrix-vector product $Ax$ and $A^{T}y$ can be computed efficiently; in particular, $A$ need not have orthogonal rows. SPA converges to the target signal by solving a sequence of penalized optimization subproblems, and each subproblem is solved using Nesterov's optimal algorithm for simple sets [Yu. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, Norwell, MA, 2004] and [Yu. Nesterov, Math. Program., 103 (2005), pp. 127-152]. We show that the SPA iterates $x_k$ are $\epsilon$-feasible; i.e. $\|Ax_k-b\|_2\leq\epsilon$ and $\epsilon$-optimal; i.e. $|~\|x_k\|_1-\|x^\ast\|_1|\leq\epsilon$ after $\tilde{\mathcal{O}}(\epsilon^{-\frac{3}{2}})$ iterations. SPA is able to work with $\ell_1$, $\ell_2$, or $\ell_{\infty}$ penalty on the infeasibility, and SPA can be easily extended to solve the relaxed recovery problem $\min\{\|x\|_1:\|Ax-b\|_2\leq\delta\}$.