2012/2013-TDA-intro-part1
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This document is an introduction to topological data analysis (TDA) presented by Rodrigo Rojas Moraleda. It discusses the principles of TDA, including that the shape of data matters and can be measured topologically by counting loops and clusters rather than with numbers. Topology provides a way to represent shapes with less information, like nodes capturing a space's structure, rather than infinite point lists. The introduction covers what topology is, elements like manifolds and simplicial complexes, and outlines subsequent parts on homology, persistent homology, and applications to point clouds.
2012/2013-TDA-intro-part1
- 1. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Introduction to Topological Data Analysis) (TDA) Part 1 - Introduction Rodrigo Rojas Moraleda April 22 2012 Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 1/31
- 2. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Outline 1 About 2 Introduction 3 Elements of Topology Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 2/31
- 3. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO 1 About 2 Introduction 3 Elements of Topology Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 3/31
- 4. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO About Image Processing Group Luis Salinas Carrasco Dr. rer. Nat (Mathematics) Professor at Department of Computer Science UTFSM. Electronic Engineer, Areas of Interest: Computer Science and Computational Methods in Engineering, Functions Theory, Theoretical Computer Science and Mathematical Foundations of Computer Science, Signal and Image Processing Steffen H¨artel Dr. rer. Nat. (Biophysics) Group leader of Laboratory of Scientific Image Processing (SCIAN-Lab) at the Medical Faculty of the University of Chile, Santiago, ChileAreas of Interest: to develop mathematical tools and computational algorithms to access dynamic, morphologic, and topologic features in experimental systems with a biophysical, biological, or medical background Rodrigo Rojas Moraleda Ph.D student in Computer Science, Engineer in Computer Science, B.Sc. Engineering minor in Computer Science. Areas of Interest: Image Processing, Computer Vision, Data Analysis, Biomedical applications Raquel Pezoa Rivera Ph.D student in Computer Science, M.Sc in Computer Science, Universidad T´ecnica Federico Santa Mar´ıa (UTFSM) Computer Science Engineer, UTFSM Paola Arce Ph.D student in Computer Science. Time series data analysis Cesar Fernandez Ph.D student in Computer Science. Time series data analysis Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 4/31
- 5. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Summary of contents Part 1 - Introduction Part 1 - Introduction Introduction & Motivation What is topology What is computational topology. Elements of Topology. Manifolds. Topological spaces. Introduction to simplicial complexes. Geometrical Simplicial Complexes. Abstract Simplicial Complexes Part 2 - Homology What is homology Elements of homology by example. Chains. Clyces. Boundary operator. Homology classes. Homology groups. Betty numbers. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 5/31
- 6. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Simplicial Homology Part 3 - Simplicial Homology Point Clouds Homotopy & Nervs Alpha complex Rips Complex Cech Complex Sandwich theorem Part 4 - Persistence homology Concepts Filtration Barcodes Uses and applications Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 6/31
- 7. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO 1 About 2 Introduction 3 Elements of Topology Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 7/31
- 8. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Priciple Shape of data matters Statistically a crudeaspect is how many connected componnets can break into. clusters groups content part of the phenomena. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 8/31
- 9. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters What is shape of data ? Formaly defined in terms of a distance funtion of a metric e.g. correlation distances binary data Hamming distances Euclidean distances alphabet letter distances in Genes sequences. Encoding notion of similarity o disimimilarity. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 9/31
- 10. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters What is shape of data ? Formaly defined in terms of a distance funtion of a metric e.g. correlation distances binary data Hamming distances Euclidean distances alphabet letter distances in Genes sequences. Encoding notion of similarity o disimimilarity. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 9/31
- 11. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters We trust in small distances Physics define a theoretical justify based on notion for distances or meassurements In life or social sciences, distance (metric) are constructed using a notion of similarity (proximity). Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 10/31
- 12. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters We trust in small distances 1.8 1.5 Both pairs are regarded as similar, but the strength of the similarity as encoded by the distance may not be so significant Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 11/31
- 13. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters We trust in small distances? Even Local Connections are Noisy, depending on observer’s scale! 15 10 5 0 -5 -10 -15 -15 -15 -10 -5 0 10 15 Is it a circle, dots, or circle of circles? To see the circle, we ignore variations in small distance (tolerance for proximity) Because distance measurements are noisy Similar objects lie in neighborhood of each other, which suffices to define topology Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 12/31
- 14. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters We trust in small distances? Even Local Connections are Noisy, depending on observer’s scale! 15 10 5 0 -5 -10 -15 -15 -15 -10 -5 0 10 15 Is it a circle, dots, or circle of circles? To see the circle, we ignore variations in small distance (tolerance for proximity) Because distance measurements are noisy Similar objects lie in neighborhood of each other, which suffices to define topology Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 12/31
- 15. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 16. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 17. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 18. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 19. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 20. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 21. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 22. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 23. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 24. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? according to Gunnar Carlsson Mathematical formalism for doing two things: ◦ Meassuring shapes · what is meassuring a shape?. · counting loops, clusters? · is not something that can be done easily in terms of a number. · is a vague notion. ◦ Representing shapes · shape in a metric spaces is a infinite list of points and distances. · we would like a way to representing and understanding the shape in a musch smaller amount of information like a triangulation or set of nodes that capture the structure of the space much simpler representation. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 13/31
- 25. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is topology? This is the subject of topology Origins of Topology in Math Leonhard Euler 1736, Seven Bridges of K¨onigsberg Johann Benedict Listing 1847, Vorstudien zur Topologie J.B. Listing (orbituary) Nature 27:316-317, 1883. “qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated.” In the last 10 years move into the field of Point Clouds. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 14/31
- 26. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters → An object representation contains enough information to reconstruct (an approximation to) the object. → A description only contains enough information to identify an object as amember of some class. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 15/31
- 27. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is Topology? According to the Oxford English Dictionary,the word topology is derived from topos (τ´oπoς) meaning place, and -logy (λoγ´ια), a variant of the verb λ´εγειυ, meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Topology - Study of shapes (topological spaces) that can be deformed into other shapes in a continuous manner (without tearing) If one shape can be deformed into another, we say they are topologically equivalent. In contrast the field of euclidean geometry studies intrinsic properties in the euclidean n-dimensional space that are invariant under euclidean transformations, such as translations and ortogonal transformations in En . Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 16/31
- 28. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO What is Computational Topology? Computational topology has theoretical and practical goals. Theoretically, looks at the tractability and complexity of each problem, as well as the design of efficient data structures and algorithms. Practically, involves heuristics and fast software for solving problems that arise in diverse disciplines. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 17/31
- 29. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Shape of data matters End Introduction. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 18/31
- 30. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO 1 About 2 Introduction 3 Elements of Topology Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 19/31
- 31. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Topology A topology on a set X is a subset T ⊆ X such that: a) ∅, X ∈ T b) Uα ∈ T ⇒ α Uα ∈ T c) U1, . . . , Un ∈ T ⇒ n k=1 Uk ∈ T A subset U ∈ T is called open set and its complement X U is called closed. The pair (X, T ) is called a topological space. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 20/31
- 32. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Topology A function f : (X, Tx ) → (Y, Ty ) is continuous if for every open set U ∈ (Y, Ty ), f −1 (U) is open in (X, Tx ), then f is called a continuous map. U (X,Tx) f -1 (U) (Y,Ty ) f Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 21/31
- 33. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Topology In a topological space (X, Tx ), the closure U of some subset set U is the intersection of all closed sets containing U (X,Tx) (X,Tx ) UU Some spanish synonyms are: adherencia, cierre. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 22/31
- 34. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Topology The interior int(U) of some U ∈ (X, Tx ) is the union of all open sets contained in U (X,Tx) U (X,Tx ) int (U) Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 23/31
- 35. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Topology The boundary ∂U of some U is ∂U = U − int(U) (X,Tx) U (X,Tx ) U (X,Tx ) int (U) (X,Tx ) U∂ Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 24/31
- 36. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Topology A limit point A is a set of {x ∈ U : ∀ U open set in (X, Tx ), x ∈ U ⇒ (U − x) U = ∅} (X,Tx) (X,Tx ) A Ux A' Spanish synonyms are: acumulaci´on. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 25/31
- 37. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Manifolds (a) 1 (b) 2 (d) 2 (e) Klein bottle(c) Toruswith ∂ Figure: Manifolds. (a) The only compact connected one-manifold is a circle S1. (b) The sphere is a two-manifold. (c) The surface of a donut, a torus, is also a two-manifold.(d) A Boy’s surface is a geometric immersion of the projective plane P2, a nonorientable two-manifold. (e) The Klein bottle is another nonorientable two-manifold. A topological space may be viewed as an abstraction of a metric space. Similarly, manifolds generalize the connectivity of d − dimensional Euclidean spaces Rd by being locally similar but globally different. A d − dimensional chart at p ∈ X is a homeomorphism ϕ : U → Rd onto an open subset of Rd , where U is a neighborhood of p and open is defined using a metric. A d-dimensional manifold (d-manifold) is a topological space X with a d − dimensional chart at every point x ∈ X. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 26/31
- 38. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes To compute information about a topological space using a computer, we need a finite representation of the space. In this section, we represent a topological space as a union of simple pieces, deriving a combinatorial description that is useful in practice. Intuitively, cell complexes are composed of Euclidean pieces glued together along seams, generalizing polyhedra. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 27/31
- 39. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes Geometrical (a) (b) (c) (d) A set of (k + 1) points {u0, · · · , uk } in R (with k ≥ 0) is called affinely independent if the k vectors (uj − u0) , with 1 ≤ j ≤ k are linearly independent. A k − simplex σ is defined as their convex hull spaned by {u0, · · · , uk } points k is called the dimension of σ and {u0, · · · , uk } the vertices of σ. Simplices in dimension 0, 1, 2, 3 are called vertices (a), edge(b), triangles(c), and tetrahedra(d), Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 28/31
- 40. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes Geometrical The simplex is the set of points defined by all convex combinations σ = k i=0 λi ui | λi > 0and k i=0 λi = 1 An l − simplex τ is called a face of a k − simplex σ if the vertices of τ are a subset of the vertices of σ. Clearly, l ≤ k holds in this case. A k − simplex has 2k+1 − 1 faces. A k-simplex has exactly 2k+1 − 1 faces. For instance, a tetrahedron consists has one face in dimension 3 (itself), four triangular faces, six edges, and four vertices which adds up to 15 = 24 − 1. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 29/31
- 41. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes Geometrical Simplicial complexes. A geometric simplicial complex K is a finite collection of simplices in Rd such that for any simplex sigma in K, every face of σ is also in K, and. for any two simplices σ, τ in K, the intersection σ ∩ τ is empty or a face of both σ and τ (and therefore,part of K as well). the dimension of K is the maximal dimension of the simplices that it contains. Note that K is a set of subsets of Rd and not a subset of Rd . K is defined as |K| = σ∈K σ Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 30/31
- 42. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes Abstract Given abstract vertices v1, · · · , vn , an abstract simplicial complex A is defined as a collection of subsets of {v1, · · · , vn}, such that with any set M ∈ A, every nonempty subset of M is also in A. The subsets of cardinality k + 1 are called abstract k − simplices of A. We can draw an abstract complex in Rd by mapping each vertex to some point in Rd and mapping a abstract k − simplex to the simplex spanned by the k + 1 corresponding points. If this drawing is a geometric simplicial complex, we call it a geometric realization of A. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 31/31
- 43. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes Abstract Given abstract vertices v1, · · · , vn , an abstract simplicial complex A is defined as a collection of subsets of {v1, · · · , vn}, such that with any set M ∈ A, every nonempty subset of M is also in A. The subsets of cardinality k + 1 are called abstract k − simplices of A. We can draw an abstract complex in Rd by mapping each vertex to some point in Rd and mapping a abstract k − simplex to the simplex spanned by the k + 1 corresponding points. If this drawing is a geometric simplicial complex, we call it a geometric realization of A. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 31/31
- 44. DEPARTAMENTO DE INFORMATICA INVESTIGACION Y POSTGRADO Topology / Simplicial complexes Abstract Given abstract vertices v1, · · · , vn , an abstract simplicial complex A is defined as a collection of subsets of {v1, · · · , vn}, such that with any set M ∈ A, every nonempty subset of M is also in A. The subsets of cardinality k + 1 are called abstract k − simplices of A. We can draw an abstract complex in Rd by mapping each vertex to some point in Rd and mapping a abstract k − simplex to the simplex spanned by the k + 1 corresponding points. If this drawing is a geometric simplicial complex, we call it a geometric realization of A. Rodrigo Rojas Moraleda — Introduction to Topological Data Analysis) (TDA) 31/31