# Topology-Preserving Dimensionality Reduction via Interleaving Optimization

arXiv: https://arxiv.org/abs/2201.13012 · HTML: https://mlsanity.com/paper/2201.13012/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 41 references that are in the map.
- First posted on arXiv 31 Jan 2022.

## Builds on (most cited first, 12 of 12)

- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Confidence sets for persistence diagrams](https://mlsanity.com/paper/1303.7117/) (2013)
- [TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data](https://mlsanity.com/paper/2009.01512/) (2020)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Homology-Preserving Dimensionality Reduction via Manifold Landmarking and Tearing](https://mlsanity.com/paper/1806.08460/) (2018)
- [PLLay: Efficient Topological Layer based on Persistence Landscapes](https://mlsanity.com/paper/2002.02778/) (2020)
- [Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence](https://mlsanity.com/paper/2106.02096/) (2021)

## Built on by (most cited first, 3 of 3)

- [Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction](https://mlsanity.com/paper/2502.20215/) (2025)
- [TopoMap++: A faster and more space efficient technique to compute projections with topological guarantees](https://mlsanity.com/paper/2409.07257/) (2024)
- [Topologically Regularized Data Embeddings](https://mlsanity.com/paper/2301.03338/) (2023)

## Abstract (arXiv)

Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the persistent homology of Vietoris-Rips filtrations can be used to identify a scale at which topological features such as clusters or holes in an embedding and original data set are in correspondence. We show how optimization seeking to minimize the interleaving distance can be incorporated into dimensionality reduction algorithms, and explicitly demonstrate its use in finding an optimal linear projection. We demonstrate the utility of this framework to data visualization.