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On the bottleneck stability of rank decompositions of multi-parameter persistence modules

Magnus Bakke Botnan, Steffen Oppermann, Steve Oudot, Luis Scoccola

TL;DR

The paper analyzes bottleneck stability for rank decompositions of multi-parameter persistence modules. It shows minimal rank decompositions by rectangles are not stable under signed matchings and introduces the rank exact decomposition derived from the rank exact structure, proving a sharp stability bound $\widehat{d_B} \le (2n-1)^2 \cdot d_I$. It also establishes the global dimension $\mathrm{gldim}^{\mathpzc{rk}}=2n-2$ and polynomial bounds on the size of the rank exact decomposition, along with a universality result for the signed bottleneck dissimilarity that precludes discriminative metrics in general. In addition, the authors prove a bottleneck stability result for hook-decomposable modules and relate hooks to rectangles, while exploring other exact structures, including the upset (limit) structure whose global dimension can diverge. Overall, the work bridges topological data analysis with representation theory of posets, providing stable, signed-barcode invariants for multi-parameter persistence and illuminating the trade-offs between stability and discriminative power.

Abstract

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

TL;DR

The paper analyzes bottleneck stability for rank decompositions of multi-parameter persistence modules. It shows minimal rank decompositions by rectangles are not stable under signed matchings and introduces the rank exact decomposition derived from the rank exact structure, proving a sharp stability bound . It also establishes the global dimension and polynomial bounds on the size of the rank exact decomposition, along with a universality result for the signed bottleneck dissimilarity that precludes discriminative metrics in general. In addition, the authors prove a bottleneck stability result for hook-decomposable modules and relate hooks to rectangles, while exploring other exact structures, including the upset (limit) structure whose global dimension can diverge. Overall, the work bridges topological data analysis with representation theory of posets, providing stable, signed-barcode invariants for multi-parameter persistence and illuminating the trade-offs between stability and discriminative power.

Abstract

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.
Paper Structure (32 sections, 50 theorems, 74 equations, 4 figures)

This paper contains 32 sections, 50 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Context and objectives
  3. Mathematical framework and contributions.
  4. Acknowledgements
  5. Background and notation
  6. Distances and dissimilarities
  7. Posets
  8. Persistence modules
  9. Persistence modules as modules
  10. Interleavings
  11. Interval modules
  12. Multisets
  13. Barcodes and signed barcodes
  14. Matchings, bottleneck distance, and signed bottleneck dissimilarity
  15. Rank decompositions
  16. ...and 17 more sections

Key Result

Lemma 2.0

There is an additive, fully faithful functor $\mathrm{Vec}^\mathscr{P} \xrightarrow{\;\;\;} \mathrm{Mod}_{\mathbb{k} \mathscr{P}}$. The functor is given by mapping $M : \mathscr{P} \xrightarrow{\;\;\;} \mathrm{Vec}$ to the module with underlying vector space $\bigoplus_{i \in \mathscr{P}} M(i)$, and

Figures (4)

  • Figure 1: The minimal rank projective resolution of \ref{['figure:example-resolution-a']} is used to define the rank exact decomposition of the $\mathscr{R}^2$-persistence module $M$, which, in this case, is an interval module. The rank exact decomposition of $M$ is shown in \ref{['figure:example-resolution-b']} by representing an indecomposable summand $\mathsf{L}_{i,j}$ by a segment joining $i$ and $j$. Summands in even homological degrees correspond to positive bars; summands in odd homological degrees correspond to negative bars. For details, see \ref{['example:rank-exact-resolution']}.
  • Figure 2: An example of a low-cost matching (right) between the rank exact decompositions of two fp$\mathscr{R}^2$-persistence modules at small interleaving distance (left). For details, see \ref{['example:stability']}.
  • Figure 3: The minimal rank decompositions of the modules $A$ and $B$ of the proof of \ref{['proposition:instability-mrd-rectangles']}, when $k=4$.
  • Figure 4: The short exact sequence in the proof of \ref{['lemma:lower-bound-bars-mrd']}, when $k = \ell = 3$.

Theorems & Definitions (103)

  • Lemma 2.0
  • Remark 2.1
  • Theorem 2.2: botnan-oppermann-oudot
  • Lemma 2.3
  • Lemma 2.4
  • Example 2.5
  • Theorem 2.6: cf. botnan-oppermann-oudot
  • Example 2.7
  • Proposition 2.7
  • Proposition 3.1
  • ...and 93 more