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.

翻译：图切割的概率松弛方法提供了一种无需特征分解的谱聚类可微替代方案，支持端到端和在线学习。然而，先前工作主要聚焦于RatioCut，缺乏通用性保证与原则性梯度。我们提出了一个统一概率框架，涵盖包括归一化切割在内的广泛切割类别。该框架通过积分表示与具有闭式前向与反向传播的高斯超几何函数，给出了期望离散切割的紧致解析上界。综合这些成果，我们为可扩展、可微的图划分提供了严谨且数值稳定的基础，覆盖了广泛的聚类与对比学习目标。