A Novel Regularized Principal Graph Learning Framework on Explicit Graph Representation

TL;DR

This paper introduces a new regularized principal graph learning framework that captures local graph structures, handles complex data with self-intersections, and guarantees convergence, improving data visualization and structure discovery.

Contribution

The paper proposes a novel framework for principal graph learning that captures local structures, learns explicit graph representations, and guarantees convergence, addressing limitations of previous methods.

Findings

Successfully learns spanning trees and weighted graphs from data.

Outperforms baseline methods on synthetic and real datasets.

Effectively uncovers underlying data structures.

Abstract

Many scientific datasets are of high dimension, and the analysis usually requires visual manipulation by retaining the most important structures of data. Principal curve is a widely used approach for this purpose. However, many existing methods work only for data with structures that are not self-intersected, which is quite restrictive for real applications. A few methods can overcome the above problem, but they either require complicated human-made rules for a specific task with lack of convergence guarantee and adaption flexibility to different tasks, or cannot obtain explicit structures of data. To address these issues, we develop a new regularized principal graph learning framework that captures the local information of the underlying graph structure based on reversed graph embedding. As showcases, models that can learn a spanning tree or a weighted undirected $\ell_1$ graph are proposed, and a new learning algorithm is developed that learns a set of principal points and a graph structure from data, simultaneously. The new algorithm is simple with guaranteed convergence. We then extend the proposed framework to deal with large-scale data. Experimental results on various synthetic and six real world datasets show that the proposed method compares favorably with baselines and can uncover the underlying structure correctly.