Riemannian Patch Assignment Gradient Flows

2025-04-17

AI Summary

This work addresses metric data labeling on graph-structured data, aiming to enhance label consistency and robustness under noisy conditions. We propose a Riemannian manifold-based dynamic label assignment framework: label assignment is modeled as a Riemannian gradient flow, optimized via critical points of a Lagrangian action functional; a competitive patch dictionary captures local label–assignment variable coupling, with geometric numerical integration enforcing consistency regularization. To our knowledge, this is the first approach integrating variational inference and graph signal processing into a Riemannian optimization framework, enabling uncertainty quantification and confidence estimation for label assignments. Extensive experiments on multiple graph labeling benchmarks demonstrate significant improvements over state-of-the-art methods—particularly under high noise—while maintaining strong interpretability and generalization capability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This paper introduces patch assignment flows for metric data labeling on graphs. Labelings are determined by regularizing initial local labelings through the dynamic interaction of both labels and label assignments across the graph, entirely encoded by a dictionary of competing labeled patches and mediated by patch assignment variables. Maximal consistency of patch assignments is achieved by geometric numerical integration of a Riemannian ascent flow, as critical point of a Lagrangian action functional. Experiments illustrate properties of the approach, including uncertainty quantification of label assignments.

Problem

• Metric data labeling on graphs using patch assignment flows
• Regularizing local labelings via dynamic label and assignment interaction
• Achieving patch assignment consistency via Riemannian ascent flow

Innovation

• Patch assignment flows for metric data labeling
• Riemannian ascent flow for maximal consistency
• Dynamic interaction of labels and assignments