# Local Linear Convergence of Forward-Backward under Partial Smoothness

arXiv: https://arxiv.org/abs/1407.5611 · HTML: https://mlsanity.com/paper/1407.5611/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2014.

## Built on by (most cited first, 13 of 13)

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- [Automatic Differentiation of Optimization Algorithms with Time-Varying Updates](https://mlsanity.com/paper/2410.15923/) (2024)
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- [Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization](https://mlsanity.com/paper/2406.15713/) (2024)
- [Dynamic Proximal Gradient Algorithms for Schatten-p Quasi-Norm Regularized Problems](https://mlsanity.com/paper/2603.00333/) (2026)

## Abstract (arXiv)

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 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.