# On the Convergence Rate of Decomposable Submodular Function Minimization

arXiv: https://arxiv.org/abs/1406.6474 · HTML: https://mlsanity.com/paper/1406.6474/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2014.

## Built on by (most cited first, 3 of 3)

- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Decomposable Submodular Function Minimization via Maximum Flow](https://mlsanity.com/paper/2103.03868/) (2021)
- [Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components](https://mlsanity.com/paper/2110.14859/) (2021)

## Abstract (arXiv)

Submodular functions describe a variety of discrete problems in machine learning, signal processing, and computer vision. However, minimizing submodular functions poses a number of algorithmic challenges. Recent work introduced an easy-to-use, parallelizable algorithm for minimizing submodular functions that decompose as the sum of "simple" submodular functions. Empirically, this algorithm performs extremely well, but no theoretical analysis was given. In this paper, we show that the algorithm converges linearly, and we provide upper and lower bounds on the rate of convergence. Our proof relies on the geometry of submodular polyhedra and draws on results from spectral graph theory.