MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization

Composite convex optimization models arise in several applications, and are especially prevalent in inverse problems with a sparsity inducing norm and in general convex optimization with simple constraints. The most widely used algorithms for convex composite models are accelerated first order methods, however they can take a large number of iterations to compute an acceptable solution for large-scale problems. In this paper we propose to speed up first order methods by taking advantage of the structure present in many applications and in image processing in particular. Our method is based on multi-level optimization methods and exploits the fact that many applications that give rise to large scale models can be modelled using varying degrees of fidelity. We use Nesterov's acceleration techniques together with the multi-level approach to achieve $\mathcal{O}(1/\sqrt{\epsilon})$ convergence rate, where $\epsilon$ denotes the desired accuracy. The proposed method has a better convergence rate than any other existing multi-level method for convex problems, and in addition has the same rate as accelerated methods, which is known to be optimal for first-order methods. Moreover, as our numerical experiments show, on large-scale face recognition problems our algorithm is several times faster than the state of the art

Successive Concave Sparsity Approximation for Compressed Sensing

In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell_1$ and $\ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted $\ell_1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.Comment: Submitted to IEEE Trans. on Signal Processin

Compressed matched filter for non-Gaussian noise

We consider estimation of a deterministic unknown parameter vector in a linear model with non-Gaussian noise. In the Gaussian case, dimensionality reduction via a linear matched filter provides a simple low dimensional sufficient statistic which can be easily communicated and/or stored for future inference. Such a statistic is usually unknown in the general non-Gaussian case. Instead, we propose a hybrid matched filter coupled with a randomized compressed sensing procedure, which together create a low dimensional statistic. We also derive a complementary algorithm for robust reconstruction given this statistic. Our recovery method is based on the fast iterative shrinkage and thresholding algorithm which is used for outlier rejection given the compressed data. We demonstrate the advantages of the proposed framework using synthetic simulations