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arXiv 1503.03703math.OC
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前向-后向型方法的活性识别和局部线性收敛

Activity Identification and Local Linear Convergence of Forward--Backward-type methods

Jingwei Liang, Jalal Fadili, Gabriel Peyré

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摘要：

本文考虑一类Forward-Backward（FB）分裂方法，它包括几个变体（如惯性格式，FISTA），用于极小化两个真凸下半连续函数之和，其中一个函数具有Lipschitz连续梯度，另一个函数相对于光滑活动流形$\mathcal{M}$是部分光滑的.我们提出了一个统一的框架，在此框架下，我们表明，这类FB型算法（i）正确地识别活动流形在有限数量的迭代（有限活动识别），和（ii）然后进入一个局部线性收敛制度，我们精确地描述了底层的活动流形的结构。对于较简单的问题，涉及多面体函数，我们有限终止。我们还建立和解释为什么FISTA（收敛序列）局部振荡，可以比FB慢。这些结果可能有许多应用，包括在信号/图像处理，稀疏恢复和机器学习。事实上，所获得的结果解释了典型的行为，已观察到的数值在这些领域的许多问题，如套索，组套索，融合套索和核规范正则化仅举几例。

英文摘要：

In this paper, we consider a class of Forward--Backward (FB) splitting methods that includes several variants (e.g. inertial schemes, FISTA) for minimizing the sum of two proper convex and lower semi-continuous functions, one of which has a Lipschitz continuous gradient, and the other is partly smooth relatively to a smooth active manifold $\mathcal{M}$. We propose a unified framework, under which we show that, this class of FB-type algorithms (i) correctly identifies the active manifolds in a finite number of iterations (finite activity identification), and (ii) then enters a local linear convergence regime, which we characterize precisely in terms of the structure of the underlying active manifolds. For simpler problems involving polyhedral functions, we show finite termination. We also establish and explain why FISTA (with convergent sequences) locally oscillates and can be slower than FB. These results may have numerous applications including in signal/image processing, sparse recovery and machine learning. Indeed, the obtained results explain the typical behaviour that has been observed numerically for many problems in these fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name only a few.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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