# A Normalized Bottleneck Distance on Persistence Diagrams and Homology Preservation under Dimension Reduction

arXiv: https://arxiv.org/abs/2306.06727 · HTML: https://mlsanity.com/paper/2306.06727/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 11 Jun 2023.

## Builds on (most cited first, 8 of 8)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [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)
- [Nonlinear Dimension Reduction via Outer Bi-Lipschitz Extensions](https://mlsanity.com/paper/1811.03591/) (2018)
- [Multidimensional Scaling: Approximation and Complexity](https://mlsanity.com/paper/2109.11505/) (2021)
- [Persistent homology for low-complexity models](https://mlsanity.com/paper/1709.01037/) (2017)
- [Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures](https://mlsanity.com/paper/2104.01672/) (2021)
- [Dimensionality Reduction for k-Distance Applied to Persistent Homology](https://mlsanity.com/paper/2110.05897/) (2021)

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

- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Dimensionality reduction for homological stability and global structure preservation](https://mlsanity.com/paper/2503.03156/) (2025)
- [A Stable Measure of Similarity for Time Series using Persistent Homology](https://mlsanity.com/paper/2501.02817/) (2025)
- [Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams](https://mlsanity.com/paper/2507.14340/) (2025)

## Abstract (arXiv)

Persistence diagrams (PDs) are used as signatures of point cloud data. Two clouds of points can be compared using the bottleneck distance dB between their PDs. A potential drawback of this pipeline is that point clouds sampled from topologically similar manifolds can have arbitrarily large dB when there is a large scaling between them. This situation is typical in dimension reduction frameworks. We define, and study properties of, a new scale-invariant distance between PDs termed normalized bottleneck distance, dN. In defining dN, we develop a broader framework called metric decomposition for comparing finite metric spaces of equal cardinality with a bijection. We utilize metric decomposition to prove a stability result for dN by deriving an explicit bound on the distortion of the bijective map. We then study two popular dimension reduction techniques, Johnson-Lindenstrauss (JL) projections and metric multidimensional scaling (mMDS), and a third class of general biLipschitz mappings. We provide new bounds on how well these dimension reduction techniques preserve homology with respect to dN. For a JL map f that transforms input X to f(X), we show that dN(dgm(X),dgm(f(X))) < e, where dgm(X) is the Vietoris-Rips PD of X, and pairwise distances are preserved by f up to the tolerance 0 < ε < 1. For mMDS, we present new bounds for dB and dN between PDs of X and its projection in terms of the eigenvalues of the covariance matrix. And for k-biLipschitz maps, we show that dN is bounded by the product of (k2-1)/k and the ratio of diameters of X and f(X). Finally, we use computational experiments to demonstrate the increased effectiveness of using the normalized bottleneck distance for clustering sets of point clouds sampled from different shapes.