Stabilization of codimension of persistence barcodes

TL;DR

This work introduces quiver codimension $Qcodim$, a barcode-based algebro-geometric statistic for persistence modules, defined as the number of interacting bar pairs in a barcode. It proves a barcode-driven formula $\text{codim}(\Omega)=\sum_{i<j} w(T_{i,j})$ with $w(T)=(m-r)(n-r)$ that mirrors quiver rank data, and shows RT-stability: for barcode-finite $V$, suitable finite approximations $V'=V|_{S'}$ preserve the codimension, making $Qcodim(V)$ effectively computable. The paper provides realization theorems demonstrating that any natural number can be realized as a quiver codimension via persistent homology of data from VR/Čech filtrations and sublevel-set filtrations. This yields a new, computable statistic linking persistent signatures to classical quiver-theoretic invariants, with potential implications for data analysis and beyond. Potential extensions include higher algebro-geometric invariants and connections to quiver polynomials and equivariant cohomology.

Abstract

Given a pointwise finite-dimensional persistence module over a totally ordered set $S$, a theorem of Crawley-Boevey guarantees the existence of a barcode. When the set $S$ is finite, the persistence module is an equioriented type-A quiver representation and the barcode identifies a distinguished point in an algebraic variety. We prove a formula for the codimension of this variety inside an ambient space of comparable representations which depends only on the combinatorics of the barcode. We therefore extend the notion of codimension to persistence modules over any set $S$ and prove that this extension is well-defined and can be effectively computed via a stabilization property of approximating quiver representations. Further, we prove realization theorems by constructing explicit examples via persistent homology of data for which the codimension can realize any natural number as its value.

Figures (4)

• Figure 1: The underlying point cloud has the form $X_c$ with $c=8$. Top: 2-skeleton of the VR complex for $s=4.5$. Bottom: 2-skeleton of the VR complex for $s=8$. Images created with the software FiltrationDemos.
• Figure 2: The $d=0$ (left) and $d=1$ (right) barcodes for $\mathop{\mathrm{PH}}\nolimits_d(\mathrm{VR}(X_8,\mathbb{R}^2))$ with $X_8$ as in Figure \ref{['fig:Rips']}. The axis labeled "time" denotes values from the indexing set $S = \mathbb{R}_{\geq0}$. Notice that the $d=0$ barcode is an example of the result of Theorem \ref{['thm:VR.cech.0']}. The images were produced using the TDA package library and GUDHI algorithm in the statistical software, R.
• Figure 3: The depicted data points form the initial $1$-cycle in $X_c$ (here with $\epsilon_1=1.3$). The red circles are boundaries of the balls used to construct the VR complex $(X_c)_{s}$ with $s=\epsilon_1 = 1.3$ on the left and $s=\sqrt{2}$ on the right. The solid black edges depict the $1$-skeleton of the VR complex; dotted lines depict edges not yet included in the VR complex. Filled gray triangles depict the 2-skeleton of the VR complex. Hence, the left picture depicts the birth of the initial $1$-cycle while the right depicts its death.
• Figure 4: Left: the sawtooth space $\Sigma_2 \subset \mathbb{R}^2$. Right: the corresponding barcode for $\mathop{\mathrm{PH}}\nolimits_0(\Sigma_2^g)$.

Theorems & Definitions (37)

• Theorem 1.1: CB, Theorem 1.1
• Example 2.1
• Example 2.2
• Example 2.3
• Theorem 3.1
• Example 3.2
• Definition 4.1
• Theorem 4.2
• proof
• Example 4.3