Thresholded Local Hyper-Flow Diffusion

2026-06-08

AI Summary

Existing localized hypergraph flow diffusion (HFD) methods struggle to preserve computational locality during iteration, leading to inefficiency. This work proposes Thresholded Local Hypergraph Flow Diffusion (TL-HFD), which maintains an active region around seed vertices and performs projected subgradient updates exclusively within this region and its boundary. New vertices are dynamically activated based on a top-k thresholding criterion, enabling gradual expansion. TL-HFD is the first method to achieve strictly local HFD updates while being provably equivalent to global updates. Theoretical analysis establishes finite-time bounds on dual suboptimality and the total activated volume. Experiments on general submodular hypergraphs demonstrate that TL-HFD matches or exceeds the clustering performance of standard HFD with significantly less activated volume, exhibiting notably enhanced robustness on noisy data.

Abstract

Local Hyper-Flow Diffusion (HFD) gives an edge-size-independent Cheeger-type guarantee for seeded clustering in general submodular hypergraphs, but existing HFD solvers do not keep intermediate computation local at every iteration. We introduce Thresholded Local HFD (TL-HFD), a first-order method that maintains an active region around the seeds, performs projected subgradient updates on that region and its immediate boundary, and expands via thresholded (top-k) boundary activation. We prove that the local update is exact: the degree-preconditioned projected subgradient step restricted to the active region and its boundary coincides with the unrestricted global update. We establish finite-time dual suboptimality for both exact and thresholded updates, treating the latter as inexact projected subgradient steps with explicit skipped-boundary error. We further derive an additive activated-volume bound controlled by realized local subgradient norms and the minimum boundary-push among newly activated vertices, and translate approximate dual optimality with localized support into a robust sweep-cut guarantee for early-stopped iterates. For general submodular cut-costs, each iteration is local in the scanned region and oracle-sensitive in the hyperedge primitive. Empirically, TL-HFD often matches or improves over HFD while activating less volume, with the largest gains on noisy instances where diffusion tends to absorb non-target vertices.

Problem

• seeded clustering
• submodular hypergraphs
• locality
• hyper-flow diffusion
• computational efficiency

Innovation

• Thresholded Local HFD
• submodular hypergraphs
• local diffusion
• projected subgradient
• sweep-cut guarantee