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Graph Cuts with Arbitrary Size Constraints Through Optimal Transport

Chakib Fettal, Lazhar Labiod, Mohamed Nadif

TL;DR

A new graph cut algorithm for partitioning graphs under arbitrary size constraints is proposed using an accelerated proximal GD algorithm which guarantees global convergence to a critical point, results in sparse solutions and only incurs an additional ratio of $\mathcal{O}(\log(n))$ compared to the classical spectral clustering algorithm but was seen to be more efficient.

Abstract

A common way of partitioning graphs is through minimum cuts. One drawback of classical minimum cut methods is that they tend to produce small groups, which is why more balanced variants such as normalized and ratio cuts have seen more success. However, we believe that with these variants, the balance constraints can be too restrictive for some applications like for clustering of imbalanced datasets, while not being restrictive enough for when searching for perfectly balanced partitions. Here, we propose a new graph cut algorithm for partitioning graphs under arbitrary size constraints. We formulate the graph cut problem as a Gromov-Wasserstein with a concave regularizer problem. We then propose to solve it using an accelerated proximal GD algorithm which guarantees global convergence to a critical point, results in sparse solutions and only incurs an additional ratio of $\mathcal{O}(\log(n))$ compared to the classical spectral clustering algorithm but was seen to be more efficient.

Graph Cuts with Arbitrary Size Constraints Through Optimal Transport

TL;DR

A new graph cut algorithm for partitioning graphs under arbitrary size constraints is proposed using an accelerated proximal GD algorithm which guarantees global convergence to a critical point, results in sparse solutions and only incurs an additional ratio of compared to the classical spectral clustering algorithm but was seen to be more efficient.

Abstract

A common way of partitioning graphs is through minimum cuts. One drawback of classical minimum cut methods is that they tend to produce small groups, which is why more balanced variants such as normalized and ratio cuts have seen more success. However, we believe that with these variants, the balance constraints can be too restrictive for some applications like for clustering of imbalanced datasets, while not being restrictive enough for when searching for perfectly balanced partitions. Here, we propose a new graph cut algorithm for partitioning graphs under arbitrary size constraints. We formulate the graph cut problem as a Gromov-Wasserstein with a concave regularizer problem. We then propose to solve it using an accelerated proximal GD algorithm which guarantees global convergence to a critical point, results in sparse solutions and only incurs an additional ratio of compared to the classical spectral clustering algorithm but was seen to be more efficient.
Paper Structure (32 sections, 3 theorems, 21 equations, 4 figures, 6 tables, 1 algorithm)

This paper contains 32 sections, 3 theorems, 21 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Balanced Clustering
  4. Constrained Clustering
  5. Gomov-Wasserstein Graph Clustering
  6. Preliminaries
  7. Graph Cuts and Spectral Clustering
  8. Minimum-cut Problem.
  9. Normalized $k$-Cut Problem.
  10. Spectral Clustering.
  11. Optimal Transport
  12. Discrete optimal transport.
  13. Discrete Gromov-Wasserstein Discrepancy.
  14. Proposed Methodology
  15. Normalized Cuts via Optimal Transport
  16. ...and 17 more sections

Key Result

Proposition 1

For step size $\alpha=\frac{1}{2\lambda}$, the iterates $\mathbf{X}^{(t)}$ generated by the nonconvex PGD algorithm for our problem are all extreme points of the transporation polytope, and as such, have at most $n+k-1$ nonzero entries.

Figures (4)

  • Figure 1: Evolution of the objective as function of the number of iterations.
  • Figure 2: Running times of Spectral clustering and OT-ncut on subsets of MNIST as a function of the number of nodes and the number of clusters.
  • Figure 3: OT-ncut and OT-rcut results on toy datasets.
  • Figure 4: Neményi post-hoc rank test results. OT-rcut and OT-ncut outperform the baselines in a statistically significant manner.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof