# Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano

arXiv: https://arxiv.org/abs/1602.07283 · HTML: https://mlsanity.com/paper/1602.07283/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2016.

## Built on by (most cited first, 10 of 10)

- [Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation](https://mlsanity.com/paper/2108.08106/) (2021)
- [Block Alternating Bregman Majorization Minimization with Extrapolation](https://mlsanity.com/paper/2107.04395/) (2021)
- [A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization](https://mlsanity.com/paper/2102.05433/) (2021)
- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)
- [A Generalization Result for Convergence in Learning-to-Optimize](https://mlsanity.com/paper/2410.07704/) (2024)
- [Using Powerful Prior Knowledge of Diffusion Model in Deep Unfolding Networks for Image Compressive Sensing](https://mlsanity.com/paper/2503.08429/) (2025)
- [Multiblock ADMM for nonsmooth nonconvex optimization with nonlinear coupling constraints](https://mlsanity.com/paper/2201.07657/) (2022)
- [On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector](https://mlsanity.com/paper/2206.13646/) (2022)
- [Convergence of the majorized PAM method with subspace correction for low-rank composite factorization model](https://mlsanity.com/paper/2406.04588/) (2024)
- [Convergent Plug-and-Play Image Restoration with Annealed Noise Levels](https://mlsanity.com/paper/2609.32393/) (2026)

## Abstract (arXiv)

An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka--Lojasiewicz (KL) inequality, which comprises a huge class of problems. Most of the recent algorithms that explicitly prove convergence using the KL inequality can cast into the abstract framework in this paper and, therefore, the generated sequence converges to a stationary point of the objective function. Additional flexibility compared to related approaches is gained by a descent property that is formulated with respect to a function that is allowed to change along the iterations, a generic distance measure, and an explicit/implicit relative error condition with respect to finite linear combinations of distance terms. As an application of the gained flexibility, the convergence of a block coordinate variable metric version of iPiano (an inertial forward--backward splitting algorithm) is proved, which performs favorably on an inpainting problem with a Mumford--Shah-like regularization from image processing.