A Theory of Sub-Barcodes

TL;DR

This work introduces sub-barcodes as a robust, data-driven surrogate for exact barcodes in persistent homology. By proving that barcode information contracts along factorizations of persistence-module morphisms (the Sub-Barcode Theorem), it provides strong guarantees even when interleavings are unavailable. The authors develop an algebraic framework linking sub-barcodes to ranks, an extension theory with Lipschitz and generalized bounds, and a discretization theory using Delaunay/Voronoi constructions that supports semi-supervised and barycentric refinements. They further embed sub-barcodes in a categorical setting, showing subobjects and presheaf perspectives for ranks, thereby offering a cohesive, geometry-aware, and category-theoretic understanding of stable topological features under partial information. The approach enables reliable inference of unknown or partially bounded barcodes from finite samples and supports practical TDA pipelines without stringent sampling assumptions.

Abstract

From the work of Bauer and Lesnick, it is known that there is no functor from the category of pointwise finite-dimensional persistence modules to the category of barcodes and overlap matchings. In this work, we introduce sub-barcodes and show that there is a functor from the category of factorizations of persistence module homomorphisms to a poset of barcodes ordered by the sub-barcode relation. Sub-barcodes and factorizations provide a looser alternative to bottleneck matchings and interleavings that can give strong guarantees in a number of settings that arise naturally in topological data analysis. The main use of sub-barcodes is to make strong claims about an unknown barcode in the absence of an interleaving. For example, given only upper and lower bounds $g\geq f\geq \ell$ of an unknown real-valued function $f$, a sub-barcode associated with $f$ can be constructed from $\ell$ and $g$ alone. We propose a theory of sub-barcodes and observe that the subobjects in the category of functors from intervals to matchings naturally correspond to sub-barcodes.

Figures (7)

• Figure 1: (Left) A contour plot of a Lipschitz function $f : X\to\mathbb{R}$. (Right) The barcode $\mathsf{Bar}(f)$ of $f$ (bottom) and a sub-barcode (top). All colors correspond to the function values indicated on the axis (bottom-right).
• Figure 2: On the left, two functions are depicted, one is an upper bound and the other is a lower bound on an unknown function $f:\mathbb{R} \to \mathbb{R}$. There is a corresponding barcode associated with the pair that matches minima in the upper bound to maxima in the lower bound. On the right is a candidate function $f$ that lies between the upper and lower bounds and its barcode $\mathsf{Bar}(f)$. The barcode of the inclusion of the upper and lower bounds is a sub-barcode of $\mathsf{Bar}(f)$.
• Figure 3: Sublevels of a 3-Lipschitz function $f : X\to\mathbb{R}$ on a subset of $\mathbb{R}^2$ (top) and its barcode $\mathsf{Bar}(f) = \mathsf{Bar}(\mathrm{H}_1\mathrm{Sub}_f)$ in dimension 1 (bottom). Colors correspond to the function values indicated on the bottom-axis.
• Figure 4: Two barcodes $A$ and $B$ and a sub-barcode matching $M : A\to B$.
• Figure 5: A function and its linear extension onto the Delaunay triangulation of a sample. The discretization introduces a spurious feature because it does not register the opening.

Theorems & Definitions (19)

• Remark 1
• Remark 2
• Example 3
• Theorem 4: The Sub-Barcode Theorem
• Remark 5
• Theorem 6
• Corollary 7
• Theorem 8
• Corollary 9
• Theorem 10