Beyond Spectral Clustering: Probabilistic Cuts for Differentiable Graph Partitioning

TL;DR

This paper introduces a unified probabilistic framework for differentiable graph partitioning that generalizes spectral clustering, providing scalable, end-to-end learning with principled gradients and theoretical guarantees.

Contribution

It extends probabilistic relaxations to a broad class of graph cuts, including Normalized Cut, with tight bounds and closed-form solutions for stable, scalable learning.

Findings

Provides tight analytic upper bounds on expected discrete cuts.

Offers closed-form forward and backward computations for gradients.

Enables scalable, end-to-end graph clustering without eigendecompositions.

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.