In this paper, we consider the Forward-Backward proximal splitting algorithm to minimize the sum of two proper closed convex functions, one of which having a Lipschitz continuous gradient and the other being partly smooth relative to an active manifold M. We propose a generic framework under which we show that the Forward-Backward (i) correctly identifies the active manifold 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.

This paper describes the training of a recurrent neural network as the letter posterior probability estimator for a hidden Markov model, off-line handwriting recognition system. The supervised training algorithm, backpropagation through time, requires target outputs to be provided for each frame. Three methods for deriving these targets are presented. A novel method based upon the forward(cid:173) backward algorithm is found to result in the recognizer with the lowest error rate.