# Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence

arXiv: https://arxiv.org/abs/2106.02096 · HTML: https://mlsanity.com/paper/2106.02096/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 53 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 2 of 2)

- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Rdimtools: An R package for Dimension Reduction and Intrinsic Dimension Estimation](https://mlsanity.com/paper/2005.11107/) (2020)

## Built on by (most cited first, 1 of 1)

- [Topology-Preserving Dimensionality Reduction via Interleaving Optimization](https://mlsanity.com/paper/2201.13012/) (2022)

## Abstract (arXiv)

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud mathbbX via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (or vCech) filtration of mathbbX to that of LmathbbX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism μ_operatornamequasi-iso or strong homotopy equivalence μ_operatornameequiv. These μ_operatornamequasi-iso and μ_operatornameequiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.