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

TL;DR

This paper introduces semi-supervised vertex hunting (SSVH), a novel approach leveraging partial barycentric coordinate information to improve simplex estimation in noisy data, with applications in network and text analysis.

Contribution

The paper develops a new semi-supervised vertex hunting method using linear algebra properties, providing theoretical error bounds and faster convergence than existing algorithms.

Findings

Achieves faster convergence rate than unsupervised methods

Provides theoretical error bounds for the proposed approach

Demonstrates effectiveness in network and text analysis applications

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.