Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction

Area of Science:

• Data Science
• Computational Topology
• Machine Learning

Background:

• High-dimensional data often contains complex cyclic patterns.
• Existing dimensionality reduction techniques may fail to preserve topological features.
• Topological Autoencoders (TopoAE) offer a promising approach but have limitations for higher dimensions.

Purpose of the Study:

• To develop a novel topology-aware dimensionality reduction method for accurate visualization of cyclic patterns.
• To address the limitations of existing methods in preserving 1-dimensional persistent homology.
• To improve the geometric reconstruction of cycles in low-dimensional embeddings.

Main Methods:

• Theoretical analysis of Topological Autoencoders (TopoAE) loss function for 0-dimensional persistent homology.
• Introduction of TopoAE++, a generalization of TopoAE for 1-dimensional persistent homology.
• Development of a cascade distortion penalty term for isometric embedding of 2-chains.
• Implementation of a fast algorithm for exact persistent homology computation on Rips filtrations.

Main Results:

• Demonstrated that zero loss in TopoAE induces identical persistence pairs for 0-dimensional persistent homology.
• Showcased the failure of naive TopoAE extensions for higher-dimensional persistent homology (d >= 1).
• TopoAE++ successfully generates cycle-aware planar embeddings with faithful geometrical reconstructions.
• Achieved improved runtimes and a better balance between topological accuracy (Wasserstein distance) and visual cycle preservation.

Conclusions:

• TopoAE++ offers a significant advancement in topology-aware dimensionality reduction for visualizing cyclic data.
• The method provides more accurate and visually faithful low-dimensional representations of complex topological structures.
• The developed algorithm and implementation contribute to the field of topological data analysis and machine learning.