# Supermodular Rank: Set Function Decomposition and Optimization

arXiv: https://arxiv.org/abs/2305.14632 · HTML: https://mlsanity.com/paper/2305.14632/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 56 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 13 of 13)

- [Guarantees for Greedy Maximization of Non-submodular Functions with Applications](https://mlsanity.com/paper/1703.02100/) (2017)
- [Optimal approximation for submodular and supermodular optimization with bounded curvature](https://mlsanity.com/paper/1311.4728/) (2013)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [Weakly Submodular Maximization Beyond Cardinality Constraints: Does Randomization Help Greedy?](https://mlsanity.com/paper/1707.04347/) (2017)
- [Optimal approximation for unconstrained non-submodular minimization](https://mlsanity.com/paper/1905.12145/) (2019)
- [Robust Maximization of Non-Submodular Objectives](https://mlsanity.com/paper/1802.07073/) (2018)
- [Expressive Power and Approximation Errors of Restricted Boltzmann Machines](https://mlsanity.com/paper/1406.3140/) (2014)
- [Hierarchical Models as Marginals of Hierarchical Models](https://mlsanity.com/paper/1508.03606/) (2015)
- [Dimension of Marginals of Kronecker Product Models](https://mlsanity.com/paper/1511.03570/) (2015)
- [Mixtures and products in two graphical models](https://mlsanity.com/paper/1709.05276/) (2017)
- [Unsupervised Metric Learning in Presence of Missing Data](https://mlsanity.com/paper/1807.07610/) (2018)
- [Project and Forget: Solving Large-Scale Metric Constrained Problems](https://mlsanity.com/paper/2005.03853/) (2020)
- [Generalized Metric Repair on Graphs](https://mlsanity.com/paper/1807.07619/) (2018)

## Abstract (arXiv)

We define the supermodular rank of a function on a lattice. This is the smallest number of terms needed to decompose it into a sum of supermodular functions. The supermodular summands are defined with respect to different partial orders. We characterize the maximum possible value of the supermodular rank and describe the functions with fixed supermodular rank. We analogously define the submodular rank. We use submodular decompositions to optimize set functions. Given a bound on the submodular rank of a set function, we formulate an algorithm that splits an optimization problem into submodular subproblems. We show that this method improves the approximation ratio guarantees of several algorithms for monotone set function maximization and ratio of set functions minimization, at a computation overhead that depends on the submodular rank.