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Change Action Derivatives in Persistent Homology

Deni Salja

TL;DR

The paper addresses the lack of a complete invariant for multiparameter persistent homology by developing a categorical calculus of finite differences. Filtrations indexed by a poset are analyzed via open-set topologies and memory/lifespan functors, culminating in a generalized rank notion for the multifiltration through a change-action derivative. The main contribution is showing that the extended homological memory functor $\mathbf{zb}F_d$ is differentiable in a categorical sense, with its derivative capturing the lifespan data encoded by the lifespan functor $\Gamma_d$. This provides a principled, incremental framework for extracting generalized persistence information (lifespans) from multi-parameter filtrations and highlights the fundamental role of change actions in organizing and computing multipersistence data.

Abstract

Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a barcode. For single parameter filtrations with homology coefficient taken in a principal ideal domain, the persistence diagram/barcode can be computed using the presentation theorem for finitely generated modules over a PID. One way to reconstruct the persistence diagram/barcode is to consider the rank of the pair group at all intervals, as defined by Edelsbrunner and Harer, which counts the number of homology classes whose lifespans are precisely said intervals respectively. In this paper we generalize the rank of the pair group for suitably `tame' filtrations, described as functors from a partially ordered set to a category of chain complexes, and show how it can be captured by a categorical version of the calculus of finite-differences for abelian groups.

Change Action Derivatives in Persistent Homology

TL;DR

The paper addresses the lack of a complete invariant for multiparameter persistent homology by developing a categorical calculus of finite differences. Filtrations indexed by a poset are analyzed via open-set topologies and memory/lifespan functors, culminating in a generalized rank notion for the multifiltration through a change-action derivative. The main contribution is showing that the extended homological memory functor is differentiable in a categorical sense, with its derivative capturing the lifespan data encoded by the lifespan functor . This provides a principled, incremental framework for extracting generalized persistence information (lifespans) from multi-parameter filtrations and highlights the fundamental role of change actions in organizing and computing multipersistence data.

Abstract

Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a barcode. For single parameter filtrations with homology coefficient taken in a principal ideal domain, the persistence diagram/barcode can be computed using the presentation theorem for finitely generated modules over a PID. One way to reconstruct the persistence diagram/barcode is to consider the rank of the pair group at all intervals, as defined by Edelsbrunner and Harer, which counts the number of homology classes whose lifespans are precisely said intervals respectively. In this paper we generalize the rank of the pair group for suitably `tame' filtrations, described as functors from a partially ordered set to a category of chain complexes, and show how it can be captured by a categorical version of the calculus of finite-differences for abelian groups.

Paper Structure

This paper contains 11 sections, 14 theorems, 58 equations, 4 figures.

Table of Contents

  1. Change Actions and Derivatives
  2. Finite Differences
  3. Multipersistence
  4. Filtrations
  5. Subobjects of Cycles
  6. Open Lattices of Up-Sets and Their Arrow Categories
  7. Cycle-Subobjects as Functors
  8. The Homological Memory Functor
  9. The Homological Lifespan Functor
  10. Change Action on Blankets
  11. Change Action Derivatives in Multipersistence

Key Result

Lemma 2.0.1

The triple ($\mathbf{Arr}(G)$, $+_\mathbf{G}$, $0_\mathbf{G}$) is a monoid in $\mathbf{Cat}$

Figures (4)

  • Figure 1: The up-set, $[z, \infty)$, generated by $z \in {\mathbb R}^2$
  • Figure 2: Two incomparable up-sets, $[x,\infty)$ and $[y, \infty)$, in $\tau$, their limit/meet in $\tau^{op}$ (pink), and their colimit/join in $\tau^{op}$ (teal).
  • Figure 3: An example of blankets in the arrow category of $\tau^{op}$.
  • Figure 4: Comparable opens may have individual blankets in $\tau^{op}$ that are incomparable

Theorems & Definitions (51)

  • Definition 1.0.2: CFTWM
  • Definition 1.0.3: Alvarez-Picallo A-P0)
  • Example 1.0.4
  • Example 1.0.5
  • Definition 1.0.6: APL
  • Example 1.0.7
  • Definition 1.0.8
  • Example 1.0.9
  • Lemma 2.0.1
  • proof
  • ...and 41 more