Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2310.07073

Pull-back Geometry of Persistent Homology Encodings

Shuang Liang·Renata Turkeš·Jiayi Li·Nina Otter·Guido Montúfar
View original source ↗Read source document ↗Export metadata ↓

Abstract

Persistent homology (PH) is a method for generating topology-inspired representations of data. Empirical studies that investigate the properties of PH, such as its sensitivity to perturbations or ability to detect a feature of interest, commonly rely on training and testing an additional model on the basis of the PH representation. To gain more intrinsic insights about PH, independently of the choice of such a model, we propose a novel methodology based on the pull-back geometry that a PH encoding induces on the data manifold. The spectrum and eigenvectors of the induced metric help to identify the most and least significant information captured by PH. Furthermore, the pull-back norm of tangent vectors provides insights about the sensitivity of PH to a given perturbation, or its potential to detect a given feature of interest, and in turn its ability to solve a given classification or regression problem. Experimentally, the insights gained through our methodology align well with the existing knowledge about PH. Moreover, we show that the pull-back norm correlates with the performance on downstream tasks, and can therefore guide the choice of a suitable PH encoding.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shuang Liang, Renata Turkeš, Jiayi Li, Nina Otter, Guido Montúfar. 2024-03-04. Pull-back Geometry of Persistent Homology Encodings. https://arxiv.org/abs/2310.07073

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

KEEP EXPLORING

Related papers

arXiv2026

New families of moment-angle manifolds diffeomorphic to connected sums of products of spheres

We prove that the moment-angle manifold $\mathcal Z_{\mathcal K}$ is diffeomorphic to a connected sum of products of spheres when $\mathcal K$ is a starshaped (in particular, polytopal) 3-dimensional simplicial sphere with exactly two missing edges that are not adjacent to each other. One of the summands of the connected sum is a product of three spheres. For neighbourly starshaped simplicial spheres $\mathcal K$ of odd dimension, we prove the diffeomorphism $\mathcal Z_{\mathcal K} \cong M_1\#\cdots\# M_k$, where each $M_i$ is a product of two spheres. We give an explicit description of the moment-angle manifolds corresponding to non-polytopal Barnette and Brückner spheres.

math.AT↗
arXiv2026

The Non-Orientable Topology of Condorcet's Paradox

Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of social choice theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened social choice theory and elucidated existing results. However, characterisations of preference cycles in topological social choice theory are lacking. In this paper, we address this gap by introducing a framework for topologically modelling preference cycles that generalises Baryshnikov's existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein bottle or real projective plane, depending on how preference cycles are represented. These findings allow us to reformulate Arrow's Impossibility Theorem in terms of the orientability of a surface as well.

math.AT↗
arXiv2026

Two-Sided Dimension Bounds for the Peterson Hit Problem via Projections and Matrix Minors

Let $\mathcal P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra over the prime field $\mathbb F_2$, viewed as an unstable module over the mod-$2$ Steenrod algebra $\mathcal A$. The well-known Peterson hit problem asks for a minimal set of generators for the $\mathcal A$-module $\mathcal P_k$. This is equivalent to determining the dimension of the cohit space $(Q\mathcal P_k)_d=(\mathcal P_k/\mathcal A^{+}\mathcal P_k)_d$, where $\mathcal A^{+}$ denotes the augmentation ideal of $\mathcal A$, for every $k\geq1$ and positive degree $d$. Although solved in every degree for at most four variables, it remains a difficult open problem in general. Furthermore, given the limitations of current tools, explicitly determining the dimension of $(Q\mathcal P_k)_d$ in the general case appears out of reach. Motivated by these limitations, we establish explicit upper and lower bounds for this dimension for arbitrary positive integers $k$ and $d.$ Our method combines binary combinatorics, linear algebra, and graph and simplicial structures associated with the generating Steenrod squares. We characterize zero rows, count zero columns, and refine rank estimates using Adem relations. Minors and zero rows of the resulting smaller matrix yield further two-sided cohit bounds without determining a complete basis or computing the full hit rank.

math.AT↗