We consider the constrained Linear Inverse Problem (LIP), where a certain atomic norm (like the $\ell_1 $ and the Nuclear norm) is minimized subject to a quadratic constraint. Typically, such cost functions are non-differentiable which makes them not amenable to the fast optimization methods existing in practice. We propose two equivalent reformulations of the constrained LIP with improved convex regularity: (i) a smooth convex minimization problem, and (ii) a strongly convex min-max problem. These problems could be solved by applying existing acceleration based convex optimization methods which provide better $ O \big( \frac{1}{k^2} \big) $ theoretical convergence guarantee. However, to fully exploit the utility of these reformulations, we also provide a novel algorithm, to which we refer as the Fast Linear Inverse Problem Solver (FLIPS), that is tailored to solve the reformulation of the LIP. We demonstrate the performance of FLIPS on the sparse coding problem arising in image processing tasks. In this setting, we observe that FLIPS consistently outperforms the Chambolle-Pock and C-SALSA algorithms--two of the current best methods in the literature.
翻译:我们考虑了受限制的线性反问题(LIP),在这个问题上,某种原子规范(如$/ell_1美元和核规范)在受四边限制的情况下被最小化(如$_1美元和核规范)。一般情况下,这种成本功能是不可区分的,因此不适应实际中存在的快速优化方法。我们建议对受限制的LIP进行两种等效的重新修改,同时改进convex常规性:(一) 顺利的粉丝最小化问题,和(二) 强烈的卷轴问题。这些问题可以通过应用基于加速的现有convex优化方法加以解决,这些方法可以提供更好的 O\bigh(\frac{1 ⁇ k ⁇ 2}\ big) 美元理论趋同保证。然而,为了充分利用这些重整方法的效用,我们还提供了一种新的算法,我们称之为快速线性反问题溶剂(FLIPS),用来解决LIP的重整。我们展示了FLIPS在图像处理任务中出现的稀少的编码问题方面的表现。我们发现,FLIPS在这种环境中,FLIPS始终超越了C-Sal-S-S-SAS-S-pol-S-S-C-S-S-S-S-S-S-SAL-C-S-S-S-S-S-SARgalgalgals-C-S-S-S-S-S-S-S-S-C-S-S-S-S-S-S-S-S-C-S-S-S-C-S-S-S-C-S-S-S-S-S-S-S-S-S-S-S-SARgalgalglass-S-C-S-S-S-S-S-S-S-C-C-C-S-S-S-S-S-S-S-S-C-C-C-C-C-C-C-C-C-C-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-