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

2025-02-27

AI Summary

Existing topological autoencoders (e.g., TopoAE) exhibit low fidelity in visualizing persistent homology in dimension one (PH¹) for high-dimensional data, as they fail to guarantee topological consistency when extended to PH¹—demonstrated by a rigorous counterexample; theoretical analysis confirms their consistency only for PH⁰. Method: We propose the first PH¹-aware topological dimensionality reduction method: it models PH¹ via the Rips filtration, introduces a cascaded distortion penalty to enforce PH¹-sensitive planar embedding, and develops an efficient C++ algorithm for exact computation of planar Rips PH¹. Results: Experiments show our method achieves the optimal trade-off between Wasserstein distance preservation and visual circular structure retention. It significantly improves geometric fidelity in reconstructing PH¹ loops and outperforms existing topology-aware dimensionality reduction methods in runtime efficiency.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Computer Vision: Other Foundations of Computer Vision
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the $0$-dimensional persistent homology (PH$^0$) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH$^d$ for $dge 1$. Based on this observation, we introduce a novel generalization of TopoAE to $1$-dimensional persistent homology (PH$^1$), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the $2$-chains filling persistent $1$-cycles, hence resulting in more faithful geometrical reconstructions of the $1$-cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.

Problem

• Accurate visualization of cyclic patterns
• Generalization to 1-dimensional persistent homology
• Fast computation of persistent homology for Rips filtrations

Innovation

• Topology-aware dimensionality reduction
• Generalization to 1-dimensional homology
• Fast PH computation algorithm