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Geometric embeddings of spaces of persistence diagrams with explicit distortions

Atish Mitra, Ziga Virk

TL;DR

The paper tackles the challenge of putting persistence diagrams into a Hilbert space to enable statistics and data analysis by constructing explicit, $1$-Lipschitz embeddings of the Bottleneck space $\mathcal{D}_n$ into $\ell^2$ and into finite-dimensional Euclidean spaces with quantified distortions. The approach centers on multi-scale, landmark-based embeddings built from covers of $\mathcal{D}_n$, each component depending on bottleneck distances to landmark diagrams, and then coherently assembling these components to achieve coarse and uniform embeddings, with explicit distortion functions. Key contributions include the general framework for multi-scale embeddings (Theorem ThmGluingScales) with illustrative distortion controls, a bounded-domain embedding with an explicit distortion bound, and a concrete injective finite-dimensional map on bounded domains. The work situates these constructions within quantitative dimension theory and discusses practical considerations, limitations (e.g., embedding multiplicity and non-injectivity at fixed scales), and avenues for future implementation and optimization to compare with existing persistence diagram vectorizations. Overall, the paper provides implementable, distortion-controlled vectorizations that bridge topological summaries and standard statistical tools, enabling more robust analyses of diagram collections in TDA.

Abstract

Let $n$ be a positive integer. We provide an explicit geometrically motivated $1$-Lipschitz map from the space of persistence diagrams on $n$ points (equipped with the Bottleneck distance) into the Hilbert space $\ell^2$. Such maps are a crucial step in topological data analysis, allowing the use of statistical methods (and thus data analysis) on collections of persistence diagrams. The main advantage of our maps as compared to most of the other such vectorizations is that they are coarse and uniform embeddings with explicit distortion functions. This allows us to control the amount of geometric information lost through their application. Furthermore, we also provide an explicit $1$-Lipschitz map from the space of persistence diagrams on $n$ points on a bounded domain into a Euclidean space with an explicit distortion function. We conclude with a differently flavored embedding of the space of persistence diagrams on $n$ points on a bounded domain into $\mathbb{R}^{n(n+1)}$. The maps we construct are fairly simple, with each component depending only on the bottleneck distance to the corresponding ``landmark" persistence diagram. Due to geometric motivation from classical dimension theory, our methods are best described as quantitative dimension theory.

Geometric embeddings of spaces of persistence diagrams with explicit distortions

TL;DR

The paper tackles the challenge of putting persistence diagrams into a Hilbert space to enable statistics and data analysis by constructing explicit, -Lipschitz embeddings of the Bottleneck space into and into finite-dimensional Euclidean spaces with quantified distortions. The approach centers on multi-scale, landmark-based embeddings built from covers of , each component depending on bottleneck distances to landmark diagrams, and then coherently assembling these components to achieve coarse and uniform embeddings, with explicit distortion functions. Key contributions include the general framework for multi-scale embeddings (Theorem ThmGluingScales) with illustrative distortion controls, a bounded-domain embedding with an explicit distortion bound, and a concrete injective finite-dimensional map on bounded domains. The work situates these constructions within quantitative dimension theory and discusses practical considerations, limitations (e.g., embedding multiplicity and non-injectivity at fixed scales), and avenues for future implementation and optimization to compare with existing persistence diagram vectorizations. Overall, the paper provides implementable, distortion-controlled vectorizations that bridge topological summaries and standard statistical tools, enabling more robust analyses of diagram collections in TDA.

Abstract

Let be a positive integer. We provide an explicit geometrically motivated -Lipschitz map from the space of persistence diagrams on points (equipped with the Bottleneck distance) into the Hilbert space . Such maps are a crucial step in topological data analysis, allowing the use of statistical methods (and thus data analysis) on collections of persistence diagrams. The main advantage of our maps as compared to most of the other such vectorizations is that they are coarse and uniform embeddings with explicit distortion functions. This allows us to control the amount of geometric information lost through their application. Furthermore, we also provide an explicit -Lipschitz map from the space of persistence diagrams on points on a bounded domain into a Euclidean space with an explicit distortion function. We conclude with a differently flavored embedding of the space of persistence diagrams on points on a bounded domain into . The maps we construct are fairly simple, with each component depending only on the bottleneck distance to the corresponding ``landmark" persistence diagram. Due to geometric motivation from classical dimension theory, our methods are best described as quantitative dimension theory.
Paper Structure (12 sections, 15 theorems, 31 equations, 14 figures)

This paper contains 12 sections, 15 theorems, 31 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistence diagrams
  4. Paths in $\mathcal{D}_{{n}}$
  5. Metric geometry
  6. Covers and maps at a single scale
  7. On $\mathcal{D}_{{1}}$
  8. On $\mathcal{D}_{{n}}$
  9. Incorporation of multiple scales: uniform and coarse embeddings
  10. Controlled quasi-embeddings on bounded domains
  11. On injectivity
  12. Conclusion

Key Result

Lemma 2.2

Fix $n \in \{1, 2, \ldots\}$. For each pair of diagrams $A, B$ there is a path $\gamma$ in $\mathcal{D}_{{n}}$ from $A$ to $B$ of length at most $2 d_{\mathcal{B}}(A, B)$.

Figures (14)

  • Figure 1: On the left are two persistence diagrams in $\mathcal{D}_{{1}}$, one given by $\bullet$, the other with $\square$. The grey squares indicate the closest points of $\Delta$. The geodesic between the two diagrams consists of two-point diagrams, one of which is indicated by the two $\circ$ points. Lemma \ref{['LemmaGeodesic']} states that if we slide the point of the first diagram to $\Delta$ first, and then slide $\Delta$ towards the point of the second diagram, we obtain a path in $\mathcal{D}_{{1}}$ of length twice the bottleneck distance between them.
  • Figure 2: $R{\mathbb G}$.
  • Figure 3: Elements of $R{\mathcal{U}}$ of Definition \ref{['DefCover1']}: open $3R/2$-balls around the points of $R{\mathbb G}$ (left) and the open $3R/2$-ball around the diagonal $\Delta$ (center). A sketch of the cover using opaque squares is given on the right. We can see that the multiplicity of this portion is $4$. The final cover is obtained by adding the ball around $\Delta$. It should be apparent that this addition does not increase the multiplicity.
  • Figure 4: Sketch for Lemma \ref{['LemSum1']}: the region where at least one of the functions ${\varphi}_{R,p}$ has value at least $R/2$ (left), and the complementary triangles (right).
  • Figure 5: Sketch for Lemma \ref{['LemSum1']}: regions where functions are at least $R/4$ using notation for $a$ and $b$ form Figure \ref{['Fig6']}.
  • ...and 9 more figures

Theorems & Definitions (46)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • ...and 36 more