Using Betweenness Centrality to Identify Manifold Shortcuts.

Area of Science:

• Machine Learning
• Data Science
• Computational Topology

Background:

• High-dimensional data poses challenges for pattern recognition and machine learning.
• Dimensionality reduction (DR) methods are crucial for making these tasks tractable by removing unwanted variance.
• Nonlinear DR methods, including ISOMAP, often depend on neighborhood graphs to calculate geodesic distances, but these graphs can contain erroneous edges.

Purpose of the Study:

• To address the topological sensitivity of neighborhood graphs in nonlinear DR methods.
• To develop a robust method for identifying and removing manifold-shorting edges in high-dimensional, noisy data.
• To integrate a novel edge-removal technique into the ISOMAP algorithm.

Main Methods:

• A divisive, edge-removal method utilizing graph betweenness centrality is proposed.
• The method is designed to robustly identify and remove edges connecting disparate manifold regions.
• The algorithm's integration into the ISOMAP workflow and its performance on graph construction in high dimensions are discussed.

Main Results:

• The proposed edge-removal method effectively identifies manifold-shorting edges.
• Integration into ISOMAP improves the robustness of geodesic distance calculations.
• Performance evaluation using Receiver Operating Characteristic (ROC) analysis on synthetic and real datasets demonstrates efficacy.

Conclusions:

• The developed graph-based edge-removal technique offers a robust solution for improving nonlinear dimensionality reduction.
• This method enhances the reliability of manifold learning algorithms when dealing with complex, high-dimensional datasets.
• The findings contribute to overcoming the challenges of handling noisy data without prior manifold knowledge.