Universality of persistence diagrams and the bottleneck and Wasserstein distances

Bubenik, Peter; Elchesen, Alex

doi:10.1016/j.comgeo.2022.101882

Cited by 9 publications

(5 citation statements)

References 17 publications

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“…It is worth emphasizing that, while a theoretical framework on metric comparison for PDs is well established (Cohen-Steiner, Edelsbrunner, and Harer 2007;Cohen-Steiner et al 2010;Bubenik and Elchesen 2022;Ali et al 2023;Chung and Lawson 2022), the PD construction already discards a lot of geometric and topological information about the datasets. The question of distinguishing what tasks are suited or not suited to be tackled through PD statistics is complex, and not fully settled.…”

Section: Related Work and Main Contributionsmentioning

confidence: 99%

“…A problem with comparison metrics between PDs is that they are computationally expensive, especially for PDs coming from H jhomology with j > 0, which are a special type of point clouds in R 2 . Computing the distance between two such PDs is treated as a matching problem between points in the plane, with stability and theoretical bounds based on the link with optimal transport distances like Wasserstein Distances (WDs) (Dobrushin 1970;Cohen-Steiner et al 2010;Panaretos and Zemel 2019;Bubenik and Elchesen 2022). Since WD computation for point clouds in dimension d ≥ 2 have O(n 3 )-complexity (Munkres 1957), where n is the number of points, PD comparison is often a bottleneck in ML data processing pipelines.…”

Section: Introductionmentioning

confidence: 99%

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A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams

Kindelan Nuñez,

Petrache,

Cerda

et al. 2024

AAAI

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Persistence diagrams (PD)s play a central role in topological data analysis, and are used in an ever increasing variety of applications. The comparison of PD data requires computing distances among large sets of PDs, with metrics which are accurate, theoretically sound, and fast to compute. Especially for denser multi-dimensional PDs, such comparison metrics are lacking. While on the one hand, Wasserstein-type distances have high accuracy and theoretical guarantees, they incur high computational cost. On the other hand, distances between vectorizations such as Persistence Statistics (PS)s have lower computational cost, but lack the accuracy guarantees and theoretical properties of a true distance over PD space. In this work we introduce a class of pseudodistances called Extended Topological Pseudodistances (ETD)s, which have tunable complexity, and can approximate Sliced and classical Wasserstein distances at the high-complexity extreme, while being computationally lighter and close to Persistence Statistics at the lower complexity extreme, and thus allow users to interpolate between the two metrics. We build theoretical comparisons to show how to fit our new distances at an intermediate level between persistence vectorizations and Wasserstein distances. We also experimentally verify that ETDs outperform PSs in terms of accuracy and outperform Wasserstein and Sliced Wasserstein distances in terms of computational complexity.

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Section: Related Work and Main Contributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams

Kindelan Nuñez,

Petrache,

Cerda

et al. 2024

AAAI

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“…. , y n )) = (d(x i , y i )) n i=1 p [8]. Furthermore the symmetric group S n acts on X n by permuting the indices and we have the quotient metric space (X n /S n , d n p ).…”

Section: Basic Properties Of Spaces Of Persistence Diagramsmentioning

confidence: 99%

“…Furthermore, such formal sums arise from empirical measures on metric spaces with a distinguished subspace. It has been shown [8] that the set of formal sums on a metric pair, which we denote D(X, A), has a canonical and universal family of metrics W p , p ∈ [1, ∞], called Wasserstein distances. Let p ∈ [1, ∞].…”

Section: Introductionmentioning

confidence: 99%

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Topological and metric properties of spaces of generalized persistence diagrams

Bubenik¹,

Hartsock²

2022

Preprint

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Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is σ-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.

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Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Bubenik

Elchesen

2022

J Appl. and Comput. Topology

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Universality of persistence diagrams and the bottleneck and Wasserstein distances

Cited by 9 publications

References 17 publications

A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams

A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams

Topological and metric properties of spaces of generalized persistence diagrams

Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

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