# Splitting methods with variable metric for KL functions

arXiv: https://arxiv.org/abs/1405.1357 · HTML: https://mlsanity.com/paper/1405.1357/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2014.

## Built on by (most cited first, 26 of 26)

- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Proximal Gradient Descent-Ascent: Variable Convergence under KL Geometry](https://mlsanity.com/paper/2102.04653/) (2021)
- [Convergence of Random Reshuffling Under The Kurdyka-Lojasiewicz Inequality](https://mlsanity.com/paper/2110.04926/) (2021)
- [A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples](https://mlsanity.com/paper/2106.02119/) (2021)
- [Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis](https://mlsanity.com/paper/2209.10825/) (2022)
- [Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization](https://mlsanity.com/paper/2212.05088/) (2022)
- [A Fast and Convergent Proximal Algorithm for Regularized Nonconvex and Nonsmooth Bi-level Optimization](https://mlsanity.com/paper/2203.16615/) (2022)
- [Convergence Analysis of Mean Shift](https://mlsanity.com/paper/2305.08463/) (2023)
- [Learning pseudo-contractive denoisers for inverse problems](https://mlsanity.com/paper/2402.05637/) (2024)
- [Gradient descent provably escapes saddle points in the training of shallow ReLU networks](https://mlsanity.com/paper/2208.02083/) (2022)
- [A Normal Map-Based Proximal Stochastic Gradient Method: Convergence and Identification Properties](https://mlsanity.com/paper/2305.05828/) (2023)
- [Hybrid ISTA: Unfolding ISTA With Convergence Guarantees Using Free-Form Deep Neural Networks](https://mlsanity.com/paper/2204.11640/) (2022)
- [Convergence Rates of Training Deep Neural Networks via Alternating Minimization Methods](https://mlsanity.com/paper/2208.14318/) (2022)
- [A New Random Reshuffling Method for Nonsmooth Nonconvex Finite-sum Optimization](https://mlsanity.com/paper/2312.01047/) (2023)
- [Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems](https://mlsanity.com/paper/2505.08909/) (2025)
- [Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods](https://mlsanity.com/paper/2301.02268/) (2023)
- [WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions](https://mlsanity.com/paper/2110.12437/) (2021)
- [Convergence Analysis of Blurring Mean Shift](https://mlsanity.com/paper/2402.15146/) (2024)
- [A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2507.01932/) (2025)
- [Stochastic Gradient Descent Revisited](https://mlsanity.com/paper/2412.06070/) (2024)
- [A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation](https://mlsanity.com/paper/2601.21467/) (2026)
- [Dynamic Proximal Gradient Algorithms for Schatten-p Quasi-Norm Regularized Problems](https://mlsanity.com/paper/2603.00333/) (2026)
- [Memory-Computation Tradeoffs in Semi Amortized Parametric Optimization](https://mlsanity.com/paper/2607.20769/) (2026)
- [A proximal subgradient method for nonconvex stochastic optimization under the Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2608.05460/) (2026)
- [Information Bottleneck under Perfect Privacy](https://mlsanity.com/paper/2608.11003/) (2026)
- [GCA: Global Centroid Alignment in Federated Learning](https://mlsanity.com/paper/2608.22593/) (2026)

## Abstract (arXiv)

We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satisfies the Kurdyka-Lojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of f and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailled.