Persistent Homology

We describe a new methodology for studying persistence of topological fea- tures across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not cov- ered by that theory. In this paper we develop theoretical and

A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

To make sense of large data sets, we often look for patterns in how data points are "shaped" in the space of possible measurement outcomes. The emerging field of topological data analysis (TDA) offers a toolkit for formalizing the process of identifying such shapes. This paper aims to discover why and how the resulting analysis should be understood as reflecting significant features of the systems that generated the data. I argue that a particular feature of TDA-its functoriality-is what enables TDA to translate visual intuitions about structure in data into precise, computationally tractable descriptions of real-world systems.

Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186-198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and metrics concerned with node and link properties, e.g. degree-distribution, node centrality and modularity. Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. This can be done by focusing on the features of a set of topological objects-homological cycles-associated with the weighted functional network. We leverage the detected topological information to define the homological scaffolds, a new set of objects designed to represent compactly the homological features of the correlation network ...

Summary. This article addresses the issue of building discrete topol ogical spaces from con- tinuous data measured on a complex system and then the statistical characterization of the obtained space. As an illustration, the sensitivity of grap hs properties to thresholding is ana- lysed. A possible way to cope with that flaw is the multilevel p oint of view. We

In this work, we suggest a collection of novel models for the representation of music. These models are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. We tackle the problem of music representation following three non-orthogonal directions. First, we propose an interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids, and their main features are visualised. Thereafter, we give a topological interpretation of the Tonnetz (a graph commonly used in computational musicology), whose vertices are deformed by both a harmonic and a consonance-oriented function. The shapes derived from these deformati...

We propose a memory-efficient method that computes persistent homology for 3D gray-scale images. The basic idea is to compute the persistence of the induced Morse-Smale complex. Since in practice this complex is much smaller than the input data, significantly less memory is required for the subsequent computations. We propose a novel algorithm that efficiently extracts the Morse-Smale complex based on

Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the Tonnetz. The Tonnetz is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the persistent properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.

Abstract: Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be ...

In Computer Vision the ability to recognize objects in the presence of occlusions is a necessary requirement for any shape representation method. In this paper we investigate how the size function of a shape changes when a portion of the shape is occluded by another shape. More precisely, considering a set X = A ∪ B and a measuring function ϕ on X, we establish a condition so that l(X,ϕ) = l(A,ϕ|A) + l(B,ϕ|B) − l(A∩B,ϕ|A∩B). The main tool we use is the Mayer-Vietoris sequence of ˇ Cech homology groups. This result allows us to prove that size functions are able to detect partial matching between shapes by showing a common subset of cornerpoints.

Abstract: Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching ...