Abstract This paper proposes a stable volume and variant, referred to as sub-volume, for more reliable data analysis using persistent homology. In prior research, an optimal cycle similar ideas have been proposed identify the homological structure corresponding each birth-death pair in persistence diagram. While this is helpful homology, results are sensitive noise. The sensitivity affects reli...

Homology is topologically invariant, meaning it is a property of an object that does not change under continuous (elastic) transformations of the object. Persistent homology studies homology classes and their life-times (persistence) in the belief that significant topological attributes must have a long life-time in a filtration (an increasing nested sequence of subcomplexes). In this demo, we ...

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudomultidimensional pe...

The Euler characteristic is an invariant of a topological space that in precise sense captures its canonical notion size, akin to the cardinality set. closely related homology space, as it can be expressed alternating sum Betti numbers, whenever well-defined. Thus, one says categorifies characteristic. In his work on generalisation cardinality-like invariants, Leinster introduced magnitude metr...

In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our is based on the observation that one can empirically simulate null distribution universal across many applications involving persistence Our suggests efficiently small number summaries collected data and use in same way p-value tables were used ...

Abstract Topological data analysis involves the statistical characterization of shape data. Persistent homology is a primary tool topological analysis, which can be used to analyze features and perform inference. In this paper, we present two-stage hypothesis test for vectorized persistence diagrams. The first stage filters vector elements in diagrams enhance power test. second consists multipl...

In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram. We start with the question of ridding the function of topological noise as suggested by its persistence diagram. We give an algorithm for hie...

We extend the results of Adcock, Carlsson, and Carlsson ([ACC13]) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson, Singh, and Zomorodian in [CSZ10]. The use of topology to study point cloud data has been well established ([Car09], [Car14]). Given a finite metric space (e.g., a finite set in R n), one first constructs a filter...

Given a function f : X → R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f . In addition, we quantify the robustness of the homology classes under perturbations of f using well groups. After characterizing these groups, we show how to read their ranks from the same ...

Persistence modules are algebraic constructs that can be used to describe the shape of an object starting from a geometric representation of it. As shape descriptors, persistence modules are not complete, that is they may not distinguish non-equivalent shapes. In this paper we show that one reason for this is that homomorphisms between persistence modules forget the geometric nature of the prob...