Bigraded Betti numbers and Generalized Persistence Diagrams

Abstract

Commutative diagrams of vector spaces and linear maps over $\mathbb{Z}^2$ are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module $M$ is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and Mémoli. Via the Möbius inversion of the generalized rank invariant of $M$, we obtain a collection of connected subsets $I\subset\mathbb{Z}^2$ with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over $\mathbb{R}$ from TDA. In this paper we show that the bigraded Betti numbers of $M$, a classical algebraic invariant of $M$, are obtained by counting the corner points of these subsets $I$s. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to $d$-parameter persistence modules for $d \geq 3$.

Figures (8)

• Figure 1: (A) A $\mathbb{Z}^2$-indexed persistence module $M$ whose support is contained in a $3\times 4$ grid. (B) $M$ is interval decomposable, and the barcode of $M$ consists of the two blue intervals of $\mathbb{Z}^2$ (Definitions \ref{['def:intervals']} and \ref{['def:barcode']}). (C) Expand each of the blue intervals from (B) to intervals in $\mathbb{R}^2$ as follows: Each point $p=(p_1,p_2)$ in the two intervals is expanded to the unit square $[p_1,p_1+1)\times[p_2,p_2+1)\subset \mathbb{R}^2$. Black dots, red stars and blue squares indicate three different corner types of the expanded intervals (see Fig. \ref{['fig:corner_points']}). The bigraded Betti numbers of $M$ can be read from these corner types; for each $p\in \mathbb{Z}^2$, $\beta_j(M)(p)$ is equal to the number of black dots, red stars, and blue squares at $p$ when $j=0,1,2$, respectively. (A') Another $\mathbb{Z}^2$-indexed persistence module $N$ whose support is contained in a $3\times 4$ grid. $N$ is not interval decomposable. (B') The $\mathbf{Int}$-generalized persistence diagram of $N$ (Definition \ref{['def:IGPD']}) is shown, where the multiplicity of the red interval is -1 and the multiplicity of each blue interval is 1. (C') is similarly interpreted as in (C), where corner points of the red interval negatively contribute to the counting of the bigraded Betti numbers. More details are provided in Example \ref{['ex:pictorial interpretation']}.
• Figure 2: An interval $I\in\mathbf{Int}(\mathbb{Z}^2)$ and its corresponding region $I^+\subset \mathbb{R}^2$ with its corner points. Points on the upper boundary (dashed lines) do not belong to $I^+$, while points on the lower boundary (solid lines) belong to $I^+$. Points which lie on both boundaries do not belong to $I^+$.
• Figure 3: Illustrations for Example \ref{['ex:dgm computation']}.
• Figure 4: The three different types of corner points in $I^+\subset \mathbb{R}^2$ and $J^+\subset \mathbb{R}^2$. Note that two different $1^\mathrm{st}$ type corner points of $J$ are located at $p$. See Definition \ref{['def:types of corners']} for a rigorous description of each of the three types of corner points.
• Figure 5: $I_1$, $I_2$, $I_3$, and $I_4$ are the intervals corresponding to Fig. \ref{['fig:introduction']} (B').

Theorems & Definitions (50)

• Remark 1.1
• Theorem 2.1: Krull-Remak-Schmidt-Azumaya azumaya1950corrections
• Definition 2.2
• Definition 2.3
• Theorem 2.4: azumaya1950correctionscrawley2015decompositiongabrielsthm
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Theorem 2.9
• Theorem 2.10: Möbius Inversion formula