Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Probabilistic Graph Cuts

Ayoub Ghriss

TL;DR

This work introduces a unified, differentiable probabilistic framework for graph cuts that extends beyond RatioCut to include Normalized Cut. By deriving tight analytic upper bounds on expected discrete cuts via integral representations and Gauss hypergeometric functions ${}_2F_1$, the authors enable stable forward/backward passes without eigendecomposition. The framework provides zero-aware AM–GM gap control, minibatch concentration guarantees, and a Hölder-product binning scheme to handle heterogeneous degrees, resulting in a scalable objective that monotonically tightens during training. Connections to SimCLR and CLIP demonstrate practical relevance to modern SSL pipelines, while the theoretical guarantees support principled optimization and potential extensions. The approach yields a practical, theory-grounded surrogate for end-to-end graph partitioning with broad implications for clustering and multimodal learning.

Abstract

Probabilistic relaxations of graph cuts offer a differentiable alternative to spectral clustering, enabling end-to-end and online learning without eigendecompositions, yet prior work centered on RatioCut and lacked general guarantees and principled gradients. We present a unified probabilistic framework that covers a wide class of cuts, including Normalized Cut. Our framework provides tight analytic upper bounds on expected discrete cuts via integral representations and Gauss hypergeometric functions with closed-form forward and backward. Together, these results deliver a rigorous, numerically stable foundation for scalable, differentiable graph partitioning covering a wide range of clustering and contrastive learning objectives.

Probabilistic Graph Cuts

TL;DR

This work introduces a unified, differentiable probabilistic framework for graph cuts that extends beyond RatioCut to include Normalized Cut. By deriving tight analytic upper bounds on expected discrete cuts via integral representations and Gauss hypergeometric functions , the authors enable stable forward/backward passes without eigendecomposition. The framework provides zero-aware AM–GM gap control, minibatch concentration guarantees, and a Hölder-product binning scheme to handle heterogeneous degrees, resulting in a scalable objective that monotonically tightens during training. Connections to SimCLR and CLIP demonstrate practical relevance to modern SSL pipelines, while the theoretical guarantees support principled optimization and potential extensions. The approach yields a practical, theory-grounded surrogate for end-to-end graph partitioning with broad implications for clustering and multimodal learning.

Abstract

Probabilistic relaxations of graph cuts offer a differentiable alternative to spectral clustering, enabling end-to-end and online learning without eigendecompositions, yet prior work centered on RatioCut and lacked general guarantees and principled gradients. We present a unified probabilistic framework that covers a wide class of cuts, including Normalized Cut. Our framework provides tight analytic upper bounds on expected discrete cuts via integral representations and Gauss hypergeometric functions with closed-form forward and backward. Together, these results deliver a rigorous, numerically stable foundation for scalable, differentiable graph partitioning covering a wide range of clustering and contrastive learning objectives.

Paper Structure

This paper contains 55 sections, 20 theorems, 137 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Probabilistic Relaxation
  4. Proposed methods
  5. The AM-GM gap
  6. Zero-aware gap control.
  7. Concentration for minibatch estimators
  8. Heterogeneous degrees
  9. Optimization objective
  10. Optimization properties
  11. Time complexity.
  12. Related Work
  13. A cross-entropy upper bound and the contrastive link
  14. SimCLR as a special case of our framework
  15. Practical recipe.
  16. ...and 40 more sections

Key Result

Proposition 2.1

For the ratio cut ($s_i\equiv 1$ for all $i\in\{1,\ldots,n\}$): where $\overline{{\bm{P}}}_{:,\ell} \vcentcolon= \tfrac{1}{n}\sum_{i=1}^n {\bm{P}}_{i\ell}$ denotes the expected fraction of vertices assigned to ${\mathbb{C}}_\ell$.

Theorems & Definitions (28)

  • Proposition 2.1: PRCut bound prcut
  • Definition 3.1: Generalized Poisson–Binomial (GPB)
  • Lemma 3.1: Integral representation
  • Lemma 3.2: Euler’s identity
  • Lemma 3.3: Properties of ${}_2F_1$
  • Theorem 3.4: Hypergeometric bound
  • Proposition 3.4: Integrated AM--GM gap
  • Corollary 3.4: Simple upper bound
  • Proposition 3.4: Zero-aware AM-GM gap
  • Proposition 3.5: Concentration of the minibatch envelope
  • ...and 18 more