Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.36843

A minimising principle for interval covers of zigzag persistence modules

Junyan Chu·Shizuo Kaji·Meiyan Kang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Zigzag persistence modules serve as an essential tool in topological data analysis. The fundamental theorem of persistent homology states that any zigzag persistence module decomposes into a direct sum of simple building blocks called interval modules. We consider an interval cover, a possibly redundant family of interval submodules spanning every vertex space of a finite zigzag persistence module. We give an elementary descent argument that replaces a redundant cover with a minimal one that is an interval decomposition. This provides an optimisation perspective on persistence modules and their interval decomposition. As an application, we give an estimate of the generalised rank invariant in terms of a cover.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Junyan Chu, Shizuo Kaji, Meiyan Kang. 2026-10-27. A minimising principle for interval covers of zigzag persistence modules. https://arxiv.org/abs/2610.36843

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Limit Computation Over Posets via Minimal Initial Functors

It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a poset. We say an initial functor $F\colon C\to D$ with $C$ small is minimal if the sets of objects and morphisms of $C$ each have minimum cardinality, among the sources of all initial functors with target $D$. For $Q$ a finite poset or $Q\subseteq \mathbb N^d$ an interval (i.e., a convex, connected subposet), we describe all minimal initial functors $F\colon P\to Q$. In particular, we show that such $F$ is always a subposet inclusion, up to isomorphism. We give efficient algorithms to compute a choice of minimal initial functor. In the case that $Q\subseteq \mathbb N^d$ is an interval, we give asymptotically optimal bounds on $|\hom P|$, the number of relations in $P$ (including identities), in terms of the number $n$ of minima of $Q$: We show that $|\hom P|=Θ(n)$ for $d\leq 3$, and $|\hom P|=Θ(n^2)$ for $d>3$. We apply these results to give new bounds on the cost of computing $\lim G$ for a functor $G \colon Q\to \mathbf{Vec}$ valued in vector spaces. For $Q$ connected, we also give new bounds on the cost of computing the generalized rank of $G$ (i.e., the rank of the induced map $\lim G\to \operatorname{colim} G$), which is of interest in topological data analysis.

math.AT↗
arXiv2026

New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance

In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural graded $\mathbb{Q}$-linear section of the homomorphism induced in cohomology by the symmetric diagonal. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to distributional homotopic distance. A sequential version of this invariant was recently introduced by Jauhari and Oprea; we give an equivalent formulation in terms of distributed homotopies and derive new fibration and multiplicative triangle inequalities.

math.AT↗
arXiv2026

On smooth and bundle structures on topological manifolds homotopy equivalent to $S^{4k-1}$-bundles over $S^{4k}$

We first verify the expected calculations of the topological structure set, in the sense of surgery theory, for the total spaces of $S^{4k-1}$-bundles over $S^{4k}$ for $k\ge3$, in which manifolds homotopy equivalent to the total spaces are organized. Next, we investigate the question of which of the elements in these structure sets can be realized as such bundles. Moreover, we study the forgetful map from the smooth version to the topological version of the structure set; we determine its image. All elements in the image turn out to have the underlying manifold such a bundle.

math.AT↗