This work considers reconstructing a target signal in a context of distributed sparse sources. We propose an efficient reconstruction algorithm with the aid of other given sources as multiple side information (SI). The proposed algorithm takes advantage of compressive sensing (CS) with SI and adaptive weights by solving a proposed weighted @math - @math minimization. The proposed algorithm computes the adaptive weights in two levels, first each individual intra-SI and then inter-SI weights are iteratively updated at every reconstructed iteration. This two-level optimization leads the proposed reconstruction algorithm with multiple SI using adaptive weights (RAMSIA) to robustly exploit the multiple SIs with different qualities. We experimentally perform our algorithm on generated sparse signals and also correlated feature histograms as multiview sparse sources from a multiview image database. The results show that RAMSIA significantly outperforms both classical CS and CS with single SI, and RAMSIA with higher number of SIs gained more than the one with smaller number of SIs.
Problem becomes finding a solution to where @math is a smooth convex function with Lipschitz constant @math @cite_5 of gradient @math and @math is a continuous convex function possibly non-smooth. Problem is obviously a special case of with @math , where @math is a regularization parameter, and @math . The results of using proximal gradient methods @cite_5 give that @math at iteration @math can be iteratively computed by: where @math and @math is a proximal operator that is defined by
@cite_5
2017938640 2551153860 2168914046 2952975724
We consider incrementally updated gradient methods for minimizing the sum of smooth functions and a convex function. This method can use a (sufficiently small) constant stepsize or, more practically, an adaptive stepsize that is decreased whenever sufficient progress is not made. We show that if the gradients of the smooth functions are Lipschitz continuous on the space of n-dimensional real column vectors or the gradients of the smooth functions are bounded and Lipschitz continuous over a certain level set and the convex function is Lipschitz continuous on its domain, then every cluster point of the iterates generated by the method is a stationary point. If in addition a local Lipschitz error bound assumption holds, then the method is linearly convergent. The proximal gradient and its variants is one of the most attractive first-order algorithm for minimizing the sum of two convex functions, with one being nonsmooth. However, it requires the differentiable part of the objective to have a Lipschitz continuous gradient, thus precluding its use in many applications. In this paper we introduce a framework which allows to circumvent the intricate question of Lipschitz continuity of gradients by using an elegant and easy to check convexity condition which captures the geometry of the constraints. This condition translates into a new descent lemma which in turn leads to a natural derivation of the proximal-gradient scheme with Bregman distances. We then identify a new notion of asymmetry measure for Bregman distances, which is central in determining the relevant step-size. These novelties allow to prove a global sublinear rate of convergence, and as a by-product, global pointwise convergence is obtained. This provides a new path to a broad spectrum of problems arising in key applications which were, until now, considered as out of reach via proximal gradient methods. We illustrate this potential by showing how our results can be applied to build new and simple schemes for Poisson inverse problems. We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates. Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems. We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.