Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! !...

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Topological Data Analysis Matteo Rucco 25/06/2015

Transcript of Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! !...

Page 1: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

S

Topological Data Analysis

Matteo Rucco 25/06/2015

Page 2: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Topology?

Page 3: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Topology

S  Topology studies shapes without using distances!

S  Topology is based on some special “functions” for classifying or acting over topological spaces

S  We will use HOMOLOGY for classifying objects!

Page 4: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Are they equivalent?

Page 5: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

They are TOPOLOGICALLY equivalent

Because you can define a continuous transformation without drilling, cutting or twisting the objects

Page 6: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Benefits

S  Topology natural provides clustering

S  Topology acts well both on small and big dataset

S  Topology is robust to noise: the holes will survive

Page 7: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Topological Data Analysis

S  Biology–Immune System dynamics

S  Medicine – Cancer detection from medical images

S  Optimization– Sensor network

Point Cloud Data – Timed Point Cloud Data – Networks

Page 8: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

DETAILS TIME

Page 9: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

S  Simplicial Complex: is a collection of simplices – higher dimensional triangle

S  Simplical complex lets you to do a finite coverage of a geometrical object

How to build a topological space

Page 10: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

How to build a Simplicial Complex

S  From Point Cloud Data – points in Rd d-dimensions: Vietoris-Rips, Cech or with Witness complexes

S  From Undirected Weighted Networks: clique-complexes

S  From Directed Weighted Networks: neighborhood-complexes

Page 11: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Vietoris-Rips Complexes

Page 12: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Clique-Complex

Page 13: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Neighborhood-Complex

Page 14: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

How to characterize a simplicial complex

S  Using HOMOLOGY. It counts the number of n-dimensional holes. Betti Number is the output of Homology and can be used for deriving equivalence classes: “clustering”

Page 15: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Persistent Homology

S  It is the implementation of homology by using an incremental filtered algorithm

Page 16: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Graphical representations of Betti Numbers

S  persistent-barcode or persistent-diagram

Page 17: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

jHoles: java tool for computing PHOM (persistent homology)

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More or less at the end…

Page 19: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Applications

S  Immune response to vaccination

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Applications

S  Time series analysis: periodicity detection

Page 21: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Applications

S  Cancer detection

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Applications

S  Phase transition of a super glass metal

Page 23: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Topology…other approaches

S  Q-Analysis

Mapper

Page 24: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Applications

S  Mapper for Pulmonary Embolism diagnosys

Page 25: Topological Data Analysis - FBK · Topology ! Topology studies shapes without using distances! ! Topology is based on some special “functions” for classifying or acting over topological

Thanks


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