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

arXiv: https://arxiv.org/abs/2502.20215 · HTML: https://mlsanity.com/paper/2502.20215/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 92 references that are in the map.
- First posted on arXiv 27 Feb 2025.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-aware Matchings](https://mlsanity.com/paper/1908.07841/) (2019)
- [TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data](https://mlsanity.com/paper/2009.01512/) (2020)
- [Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data -- An Algorithm and A Benchmark](https://mlsanity.com/paper/2206.13932/) (2022)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Topological data analysis of vortices in the magnetically-induced current density in LiH molecule](https://mlsanity.com/paper/2212.08690/) (2022)
- [Homology-Preserving Dimensionality Reduction via Manifold Landmarking and Tearing](https://mlsanity.com/paper/1806.08460/) (2018)
- [Topology-Preserving Dimensionality Reduction via Interleaving Optimization](https://mlsanity.com/paper/2201.13012/) (2022)
- [TopoMap++: A faster and more space efficient technique to compute projections with topological guarantees](https://mlsanity.com/paper/2409.07257/) (2024)

## Built on by (most cited first, 4 of 4)

- [Torsion in Persistent Homology and Neural Networks](https://mlsanity.com/paper/2506.03049/) (2025)
- [Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases](https://mlsanity.com/paper/2602.22873/) (2026)
- [Understanding Differentiable Embeddings Through Differential and Integral Geometry](https://mlsanity.com/paper/2608.06809/) (2026)
- [Operational Abstractions of Neural Network Concepts via Topological Representations](https://mlsanity.com/paper/2610.05017/) (2026)

## Abstract (arXiv)

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 (PH0) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PHd for d≥ 1. Based on this observation, we introduce a novel generalization of TopoAE to 1-dimensional persistent homology (PH1), 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.