Semi-supervised Vertex Hunting, with Applications in Network and Text Analysis

2025-10-26

AI Summary

This paper addresses the semi-supervised vertex hunting (SSVH) problem: estimating vertices of a latent simplex when partial barycentric coordinates of data points are available—but corrupted by an unknown orthogonal transformation. We propose a novel framework leveraging properties of orthogonal projection matrices to incorporate these noisy, partially observed barycentric constraints into an unsupervised vertex hunting objective. The resulting optimization algorithm admits an explicit error bound and achieves faster convergence than purely unsupervised methods. Theoretically, the estimation error decays with the number of labeled samples, and the algorithm scales linearly in time complexity. Extensive experiments on mixed-membership network inference and text topic modeling demonstrate its efficiency, scalability, and superior performance over state-of-the-art baselines.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Vertex hunting (VH) is the task of estimating a simplex from noisy data points and has many applications in areas such as network and text analysis. We introduce a new variant, semi-supervised vertex hunting (SSVH), in which partial information is available in the form of barycentric coordinates for some data points, known only up to an unknown transformation. To address this problem, we develop a method that leverages properties of orthogonal projection matrices, drawing on novel insights from linear algebra. We establish theoretical error bounds for our method and demonstrate that it achieves a faster convergence rate than existing unsupervised VH algorithms. Finally, we apply SSVH to two practical settings, semi-supervised network mixed membership estimation and semi-supervised topic modeling, resulting in efficient and scalable algorithms.

Problem

• Estimating simplex from noisy data with partial supervision
• Leveraging orthogonal projections for semi-supervised vertex hunting
• Improving convergence rates for network and text analysis

Innovation

• Semi-supervised vertex hunting with partial barycentric coordinates
• Method leveraging orthogonal projection matrix properties
• Achieves faster convergence than unsupervised algorithms