In this paper, we consider the Forward--Backward proximal splitting algorithm
to minimize the sum of two proper convex functions, one of which having a
Lipschitz continuous gradient and the other being partly smooth relative to an
active manifold $\mathcal{M}$. We propose a generic framework under which we
show that the Forward--Backward (i) correctly identifies the active manifold
$\mathcal{M}$ in a finite number of iterations, and then (ii) enters a local
linear convergence regime that we characterize precisely. This gives a grounded
and unified explanation to the typical behaviour that has been observed
numerically for many problems encompassed in our framework, including the
Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to
name a few. These results may have numerous applications including in
signal/image processing processing, sparse recovery and machine learning.

本文介绍了一种前向 - 后向投影算法，用来最小化由两个凸函数的和组成的式子，其中一个具有 Lipschitz 连续梯度，另一个相对于主动流形部分平滑，同时给出算法迭代过程中的两个结果。结果表明该算法在包括 Lasso、group Lasso、fused Lasso 和核范数正则化等多个问题中都具有较广泛的应用价值。