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Topological Data Analysis

Matteo Rucco 25/06/2015

Topology?

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!

Are they equivalent?

They are TOPOLOGICALLY equivalent

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

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

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

DETAILS TIME

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

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

Vietoris-Rips Complexes

Clique-Complex

Neighborhood-Complex

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”

Persistent Homology

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

Graphical representations of Betti Numbers

S  persistent-barcode or persistent-diagram

jHoles: java tool for computing PHOM (persistent homology)

More or less at the end…

Applications

S  Immune response to vaccination

Applications

S  Time series analysis: periodicity detection

Applications

S  Cancer detection

Applications

S  Phase transition of a super glass metal

Topology…other approaches

S  Q-Analysis

Mapper

Applications

S  Mapper for Pulmonary Embolism diagnosys

Thanks