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Differential Calculus and Optimization in Persistence Module Categories

Steve Oudot

Abstract

Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other areas of mathematics, including symplectic topology, complex analysis, and topological data analysis, where they arise from filtrations of topological spaces by the sublevel sets of real-valued functions. Two fundamental properties of persistence modules make them useful in such contexts: (1) the fact that they are stable under perturbations of the originating functions, and (2) the fact that they can be approximated, in the sense of relative homological algebra, by classes of indecomposable modules with an elementary structure. In this text we give an introduction to the theory of persistence modules, then we explain how the above properties can be leveraged to build a framework for differential calculus and optimization with convergence guarantees in persistence module categories.

Differential Calculus and Optimization in Persistence Module Categories

Abstract

Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other areas of mathematics, including symplectic topology, complex analysis, and topological data analysis, where they arise from filtrations of topological spaces by the sublevel sets of real-valued functions. Two fundamental properties of persistence modules make them useful in such contexts: (1) the fact that they are stable under perturbations of the originating functions, and (2) the fact that they can be approximated, in the sense of relative homological algebra, by classes of indecomposable modules with an elementary structure. In this text we give an introduction to the theory of persistence modules, then we explain how the above properties can be leveraged to build a framework for differential calculus and optimization with convergence guarantees in persistence module categories.

Paper Structure

This paper contains 16 sections, 11 theorems, 55 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Persistence Modules
  3. Sublevel-sets topology
  4. Structure of persistence modules
  5. Direct-sum decompositions
  6. Homological approximations
  7. Stability of persistence modules
  8. Interleaving distance
  9. Bottleneck distance
  10. Signed bottleneck distance
  11. Differential calculus
  12. Setup
  13. Results
  14. Examples
  15. Optimization
  16. ...and 1 more sections

Key Result

Theorem 4.3

Suppose $(P, \leq)$ is totally ordered. Then, any $M\in \mathsf{vect}_{\mathbb{k}}^P$ decomposes as follows: where each $I_j$ is an interval of $P$. This decomposition is unique up to isomorphism and reordering of the terms.

Figures (10)

  • Figure 1: The $t=(t_1, t_2)$-sublevel set of the function $f=(f_1, f_2)\colon X\to\mathbb{R}^2$ defined by the orthogonal projections onto the coordinate axes.
  • Figure 2: Filtration of the distance function (excerpt with $10$ different levels $t$, starting at $t=0$).
  • Figure 3: Bi-filtration of the distance function and co-density estimator (excerpt with $10$ different levels for the distance and $7$ for the co-density). The distance levels are represented horizontally, the co-density levels vertically. The indices highlighted by thick rectangles arranged in a $3\times 3$ grid exhibit three connected components that merge two-by-two in three different ways then all together.
  • Figure 4: The topological data analysis pipeline.
  • Figure 5: The signed barcode $\mathrm{Bar}_{\mathcal{I}}(M)$ of the indecomposable module $M$ of Example \ref{['ex:hook-resol']}, represented as an actual barcode in the Euclidean plane, where, by convention, each interval $[p,\infty) \setminus [q,\infty)$ gives rise to a copy of the line segment $[p,q)$, while each interval $[p,\infty)$ gives rise to a copy of the diagonal ray starting at $p$. The bars in the barcode are colored according to their sign: blue for positive, red for negative.
  • ...and 5 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Example 3.1
  • Example 3.2
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3: botnan-crawleyboveyCrawley-Boevey2012
  • Example 4.4
  • Definition 4.5
  • Example 4.6
  • Theorem 4.7: botnan-crawleybovey
  • ...and 18 more