Fast Queries of Fibered Barcodes

TL;DR

The paper develops an updated, streamlined presentation of the augmented arrangement framework for fast fibered barcode queries in bipersistence. By using minimal presentations as input, the authors simplify the main algorithm and derive explicit size and time bounds for constructing and querying the augmented arrangement $\mathcal{A}^\bullet(M)$, as well as the barcode templates across its faces. The approach enables real-time querying of the fibered barcode $\mathcal{F}(M)$ via push maps and a face-dependent template library, with query times near logarithmic in the arrangement size. This work enhances the practicality of RIVET for interactive visualization and analysis of multiparameter persistence while providing concrete complexity guarantees. The results bridge algebraic foundations with geometric data structures (line arrangements and point-location) to support scalable, real-time exploration of 2-parameter persistent homology invariants.

Abstract

The fibered barcode $\mathcal{F}(M)$ of a bipersistence module $M$ is the map sending each non-negatively sloped affine line $\ell \subset \mathbb{R}^2$ to the barcode of the restriction of $M$ along $\ell$. The simplicity, computability, and stability of $\mathcal{F}(M)$ make it a natural choice of invariant for data analysis applications. In an earlier preprint [arXiv:1512.00180], we introduced a framework for real-time interactive visualization of $\mathcal{F}(M)$, which allows the user to select a single line $\ell$ via a GUI and then plots the associated barcode. This visualization is a key feature of our software RIVET for the visualization and analysis of bipersistent homology. Such interactive visualization requires a framework for efficient queries of $\mathcal{F}(M)$, i.e., for quickly obtaining the barcode along a given line $\ell$. To enable such queries, we introduced a novel data structure based on planar line arrangements, called an augmented arrangement. The aim of the present paper is to give an updated and improved exposition of the parts of our preprint [arXiv:1512.00180] concerning the mathematics of the augmented arrangement and its computation. Notably, by taking the input to be a minimal presentation rather than a chain complex, we are able to substantially simplify our main algorithm and its complexity analysis.

Figures (10)

• Figure 1: The barcode $\mathcal{B}$ is obtained from the barcode template $\mathcal{B}^\Delta$ by "pushing" the points of each pair onto $\ell$. In this example, $\mathcal{B}^\Delta=\{(a_1,b_1),(a_2,b_2)\}$, and $\mathcal{B}$ consists of two disjoint intervals $I_1$, $I_2$.
• Figure 2: Illustration of point-line duality
• Figure 3: Illustration of $\mathop{\mathrm{push}}\nolimits_{\ell}$ for a line $\ell$ of positive, finite slope.
• Figure 4: Illustration of a totally ordered partition $S^\ell$ of $S$. The elements of $S$ are drawn as black dots.
• Figure 5: An illustration of how incomparable points $s,t \in S$ push onto lines $\ell$ and $\ell'$ lying above and below the anchor $s \mathop{\mathrm{\vee}}\nolimits t$. The dual points $\ell^*$ and $\ell'^*$ lie in neighboring $2$-cells of $\mathcal{A}(M)$ with shared boundary lying on $(s \mathop{\mathrm{\vee}}\nolimits t)^*$. Note that $\mathop{\mathrm{push}}\nolimits_{\ell}(s)<\mathop{\mathrm{push}}\nolimits_{\ell}(t)$ and $\mathop{\mathrm{push}}\nolimits_{\ell'}(t)<\mathop{\mathrm{push}}\nolimits_{\ell'}(s)$, i.e., the order in which the points push onto the line switches as the line changes.

Theorems & Definitions (36)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1: cohen2006vines
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Definition 3.4
• Lemma 3.5