Barcodes: the Persistent Topology of Data
نویسندگان
چکیده
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets—persistent homology—and a novel representation of this algebraic characterization— barcodes. We sketch an application of these techniques to the classification
منابع مشابه
Cat 2015 Topological Data Analysis: New Developments and Challenges
s _______________________________________________________________________________________ Ulrich Bauer Title:Induced Matchings and the Algebraic Stability of persistence Barcode Abstract: We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the...
متن کاملVariable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally persistence, is a central tool in topological data analysis, which examines the structure of data through topology. The basic techniques have been extended in several different directions, encoding topological features by so called b...
متن کاملTopological Data Analysis
This paper explores the new and growing field of topological data analysis (TDA). TDA is a data analysis method that provides information about the ’shape’ of data. The paper describes what types of shapes TDA detects and why these shapes having meaning. Additionally, concepts from algebraic topology, the mathematics behind TDA, will be discussed. Specifically, the concepts of persistent homolo...
متن کاملHomological analysis of multi-qubit entanglement
In the last few decades the interest on quantum entanglement has been turned from purely foundational/philosophical aspects to more practical/applicative ones, thanks to quantum information processing. Along this line entanglement characterization becomes of uppermost importance, although it results a daunting task in multi-partite systems. Many approaches have been developed, based on combinat...
متن کاملPersistence Diagrams as Diagrams: A Categorification of the Stability Theorem
Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jabłoński, and Mrozek suggests an equivalent description of barcodes as functors R → Mch, where R is the poset category of real numbers and Mch is the category whose obje...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006