Persistent topology for natural data analysis - A survey

Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and classification of cells, lesions, music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brain connections, languages, handwritten and gestured letters are shown

On the geometrical properties of the coherent matching distance in 2D persistent homology

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.Comment: 39 pages, 15 figures. Corrected the definition of multiplicity of points in the extended Pareto grid and the definition of normal function. Removed Rem. 3.3. Added Ex. 3.9, Fig. 11, Fig. 12, Rem. 5.3 and Fig. 15. Changed Rem. 4.9 into regular text. Reformulated statements of Theorems 5.1, 5.2, 5.4. Some changes in their proofs. Added references. Some small changes in the text and in the figure

A Brief Introduction to Multidimensional Persistent Betti Numbers

In this paper, we propose a brief overview about multidimensional persistent Betti numbers (PBNs) and the metric that is usually used to compare them, i.e., the multidimensional matching distance. We recall the main definitions and results, mainly focusing on the 2-dimensional case. An algorithm to approximate n-dimensional PBNs with arbitrary precision is described

BPS Spectra, Barcodes and Walls

BPS spectra give important insights into the non-perturbative regimes of supersymmetric theories. Often from the study of BPS states one can infer properties of the geometrical or algebraic structures underlying such theories. In this paper we approach this problem from the perspective of persistent homology. Persistent homology is at the base of topological data analysis, which aims at extracting topological features out of a set of points. We use these techniques to investigate the topological properties which characterize the spectra of several supersymmetric models in field and string theory. We discuss how such features change upon crossing walls of marginal stability in a few examples. Then we look at the topological properties of the distributions of BPS invariants in string compactifications on compact threefolds, used to engineer black hole microstates. Finally we discuss the interplay between persistent homology and modularity by considering certain number theoretical functions used to count dyons in string compactifications and by studying equivariant elliptic genera in the context of the Mathieu moonshine

Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards

I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set $\{f_p: \mathcal{K}^p \to \mathbb{R}\}_{p \in B}$ of filtrations that is parameterized by a topological space $B$). Special cases include vineyards, the persistent homology transform, and fibered barcodes for multiparameter persistence modules. I prove that if $B$ is a smooth compact manifold, then for a generic fibered filtration function, $B$ is stratified such that within each stratum $Y \subseteq B$, there is a single PD "template" (a list of "birth" and "death" simplices) that can be used to obtain the PD for the filtration $f_p$ for any $p \in Y$. If $B$ is compact, then there are finitely many strata, so the PD bundle for a generic fibered filtration on $B$ is determined by the persistent homology at finitely many points in $B$. I also show that not every local section can be extended to a global section (a continuous map $s$ from $B$ to the total space $E$ of PDs such that $s(p) \in \textrm{PD}(f_p)$ for all $p \in B$). Consequently, a PD bundle is not necessarily the union of "vines" $\gamma: B \to E$; this is unlike a vineyard. When there is a stratification as described above, I construct a cellular sheaf that stores sufficient data to construct sections and determine whether a given local section can be extended to a global section.Comment: 40 pages, 8 figures. Substantial mathematical additions and expository changes throughou

Persistent Homology and String Vacua

We use methods from topological data analysis to study the topological features of certain distributions of string vacua. Topological data analysis is a multi-scale approach used to analyze the topological features of a dataset by identifying which homological characteristics persist over a long range of scales. We apply these techniques in several contexts. We analyze N=2 vacua by focusing on certain distributions of Calabi-Yau varieties and Landau-Ginzburg models. We then turn to flux compactifications and discuss how we can use topological data analysis to extract physical informations. Finally we apply these techniques to certain phenomenologically realistic heterotic models. We discuss the possibility of characterizing string vacua using the topological properties of their distributions.Comment: 32 pages, 12 pdf figure

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Given a normally hyperbolic invariant manifold Λ for a map f , whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future. We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map is symplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions. We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type. We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (1) (2006) 64–188] using methods of fast–slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.Peer Reviewe