Computational Topology - Mapper Jiaqi Ni Eindhoven University of - - PowerPoint PPT Presentation

Computational Topology - Mapper

Jiaqi Ni

Eindhoven University of Technology

June 14, 2018

Outline

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Introduction

◮ Mapper is a computational method for extracting simple

descriptions of high dimensional data sets in the form of simplicial complexes.

Recap about Reeb Graph

Definition: The Reeb graph of f is the set of contours R(f).

Recap about Reeb Graph

We can get similar result as Reeb Graph with Mapper.

Recap about Reeb Graph

We can also get the more different results from Reeb Graph with Mapper.

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Cover of space

If the set X is a topological space, then a cover C of X is a collection of subsets U of X whose union is the whole space

• X. In this case we say that C covers X, or that the sets U

cover X.

Cover of space

If the set X is a topological space, then a cover C of X is a collection of subsets U of X whose union is the whole space

• X. In this case we say that C covers X, or that the sets U

cover X. Topological Space X Cover of Space X

Cover of space

If Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if Y ⊆

• α∈C

Uα

Cover of space

If Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if Y ⊆

• α∈C

Uα

Cover refinement

◮ A refinement of a cover C of a topological space X is a new

cover D of X such that every set in D is contained in some set in C.

Cover refinement

◮ A refinement of a cover C of a topological space X is a new

cover D of X such that every set in D is contained in some set in C.

◮ Formally: D = {Vβ∈B} is a refinement of C = {Uα∈A}

when ∀β ∃α Vβ ⊆ Uα

Cover refinement

◮ A refinement of a cover C of a topological space X is a new

cover D of X such that every set in D is contained in some set in C.

◮ Formally: D = {Vβ∈B} is a refinement of C = {Uα∈A}

when ∀β ∃α Vβ ⊆ Uα Space X Cover of Space X Refinement of Cover

Mapper in the continuous setting

Input:

◮ Continuous function(filter) f : X → R ◮ Cover C of im(f) by open intervals: im(f ) ⊆

• c∈C

c Method:

◮ Compute pullback cover U of X: U = f −1(c)c∈C ◮ Refine U by separating each of its elements into its various

connected components → connected cover V

◮ The Mapper is the nerve of V:

◮ 1 vertex per element V ∈ V ◮ 1 edge per intersection V ∪ V ′ = ø, V , V ′ ∈ V ◮ 1 k-simplex per (k + 1)-fold intersection,

k

i=0 Vi = ø, V0, V1...Vk ∈ V

Example of Mapper in the continuous setting

Example of Mapper in the continuous setting

Example of Mapper in the continuous setting

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Mapper in practice

Input:

◮ Point cloud P with distance matrix ◮ Continuous function(filter) f : P → R ◮ Cover C of im(f) by open intervals: im(f ) ⊆

• c∈C

c Method:

◮ Compute pullback cover U of X: U = f −1(c)c∈C ◮ Refine U by applying clustering algorithm(with distance

threshold δ) → connected cover V

◮ The Mapper is the nerve of V:

◮ 1 vertex per element V ∈ V ◮ 1 edge per intersection V ∪ V ′ = ø, V , V ′ ∈ V ◮ 1 k-simplex per (k + 1)-fold intersection,

k

i=0 Vi = ø, V0, V1...Vk ∈ V

Example of Mapper in practice

Example of Mapper in practice

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Parameters of Mapper in practice

◮ Filter f : P → R

Parameters of Mapper in practice

◮ Filter f : P → R ◮ Cover C of im(f) by open intervals:

Parameters of Mapper in practice

◮ Filter f : P → R ◮ Cover C of im(f) by open intervals: ◮ Clustering algorithm

Parameters of Mapper in practice - Filter functions

◮ The outcome of Mapper is highly dependent on the function

chosen to partition (filter) the data set and the choice of functions depends mostly on the dataset.

◮ Possible functions:

◮ Density ◮ Eccentricity ◮ Graph Laplacians ◮ sum/average/max/min ◮ x/y- axis projection

Filter function examples

Filter function examples

Filter function examples

Filter function examples

Parameters of Mapper in practice - Cover

◮ Uniform cover I

◮ resolution / granularity: r (diameter of intervals) ◮ gain: g (percentage of overlap)

Parameters of Mapper in practice - Cover

◮ Uniform cover I

◮ resolution / granularity: r (diameter of intervals) ◮ gain: g (percentage of overlap)

◮ Example:

Parameters of Mapper in practice - Cover

◮ Uniform cover I

◮ resolution / granularity: r (diameter of intervals) ◮ gain: g (percentage of overlap)

◮ Example: ◮ Modification of r and g can highly effect the result.

Cover examples

Cover examples

Cover examples

Cover examples

Mapper for Y-shape point cloud data

Mapper for Y-shape point cloud data

Parameters of uniform Cover

Parameter r:

◮ Small r: fine cover, Mapper close to Reeb Graph, but

sensitive to δ.

◮ Large r: rough cover, less sensitive to δ, but Mapper far from

Reeb Graph. Parameter g:

◮ Large g(close to 1): more points inside intersections, less

sensitive to δ but far from Reeb Graph.

◮ Small g(close to 0): controlled Mapper dimension, close to

Reeb Graph.

Parameters of Mapper in practice - Clustering algorithm

Single-linkage clustering is one of several methods of hierarchical clustering.

◮ Based on grouping clusters in bottom-up fashion

(agglomerative clustering).

◮ At each step combining two clusters that contain the closest

pair of elements not yet belonging to the same cluster as each

• ther.

Example of Single-linkage clustering

Example of Single-linkage clustering

Example of Single-linkage clustering

Example of Single-linkage clustering

Example of Single-linkage clustering

Example of Single-linkage clustering

Example of Single-linkage clustering

Example of Clustering algorithm with different parameters

Example of Clustering algorithm with different parameters

Example of Clustering algorithm with different parameters

Example of Clustering algorithm with different parameters

Parameters of graph neighborhood size

Parameter δ:

◮ Large δ: fewer nodes, clean Mapper but far from Reeb

Graph(more straight lines).

◮ Small δ: presence of topological structure but lots of nodes

(noisy).

Higher Dimensional Parameter Spaces

◮ We use 1 function and let R to be our 1-dimensional

parameter space.

Higher Dimensional Parameter Spaces

◮ We use 1 function and let R to be our 1-dimensional

parameter space.

◮ We can use M functions and let RM to be our M-dimensional

parameter space, remain to find a covering of an M-dimensional hypercube which is defined by the ranges of the M functions.

Example of parameter space R2

◮ Assume we have a point could dataset P (2-Dim) as following. ◮ Assume we have two filter functions f : P → R, g : P → R,

and f = f −1 and g = g−1.

Example of parameter space R2

◮ Moreover, assume we have the following cover C, which is

also the cover of P since f = f −1 and g = g−1.

Example of parameter space R2

◮ Moreover, assume we have the following cover C, which is

also the cover of P since f = f −1 and g = g−1.

◮ Assume the clustering algorithm group every points in each

rectangle as one cluster.

Example of parameter space R2

◮ Moreover, assume we have the following cover C, which is

also the cover of P since f = f −1 and g = g−1.

◮ Assume the clustering algorithm group every points in each

rectangle as one cluster.

Example of parameter space R2

◮ Moreover, assume we have the following cover C, which is

also the cover of P since f = f −1 and g = g−1.

◮ Assume the clustering algorithm group every points in each

rectangle as one cluster.

◮ Whenever clusters corresponding to any n vertices have non

empty intersection, add a corresponding n-1 simplex.

Example of parameter space R2

◮ Two clusters intersection = 1 edge.

Example of parameter space R2

◮ Three clusters intersection = 1 triangle.

Example of parameter space R2

◮ Four clusters intersection = 1 tetrahedron.

Example of parameter space R2

◮ Final simplical complex.

Higher Dimensional Parameter Spaces

Mapper to the parameter space RM can be extended in a similar fashion (by finding a covering of an M-dimensional hypercube which is defined by the ranges of the M functions).

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Mapper in Applications

Most commonly used in:

◮ Clustering ◮ Feature selection (flares, loops)

Applications to Medical science data

145 patients who had diabetes, for each patient, six quantities were measured:

◮ Age ◮ Relative weight ◮ Fasting plasma glucose ◮ Area under the plasma glucose curve for the three hour

glucose tolerance test (OGTT)

◮ Aarea under the plasma insulin curve for the (OGTT) ◮ Steady state plasma glucose response

This creates a 6 dimensional data set.

Applications to Medical science data

◮ Applying projection pursuit methods to obtain a projection

into three dimensional Euclidean space

Applications to Medical science data

◮ Applying projection pursuit methods to obtain a projection

into three dimensional Euclidean space We want to use Mapper as an automatic tool for detecting such flares in the data.

Applications to Medical science data

◮ Left: 3 intervals, 50% overlap. ◮ Right: 4 intervals, 50% overlap. ◮ For each output:

◮ Left flare: adult onset Right flare: juvenile onset ◮ Distance function: L2-distance ◮ Filter function: density kernel with e=130,000

Mapper in Applications

◮ Innate and adaptive T cells in asthmatic patients:

Relationship to severity and disease mechanisms, Hinks et al.,

• J. Allergy Clinical Immunology, 2015

◮ Topological Data Analysis for Discovery in Preclinical Spinal

Cord Injury and Traumatic Brain Injury, Nielson et al., Nature, 2015

◮ Using Topological Data Analysis for Diagnosis Pulmonary

Embolism, Rucco et al., arXiv preprint, 2014

◮ CD8 T-cell reactivity to islet antigens is unique to type 1

while CD4 T-cell reactivity exists in both type 1 and type 2 diabetes, Sarikonda et al., J. Autoimmunity, 2013

◮ Extracting insights from the shape of complex data using

topology, Lum et al., Nature, 2013

◮ Topological Methods for Exploring Low-density States in

Biomolecular Folding Pathways, Yao et al., J. Chemical Physics, 2009

Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary

Summary

◮ Mapper: a computational method which retrieves a

higher-level understanding of the structure of data.

◮ Mapper in continuous setting. ◮ Mapper in practice ◮ Parameters of Mapper in practice

◮ filter function. ◮ covering algorithm. ◮ clustering algorithm.

◮ Applications

Sources

◮ [SMG07] G. Singh, F. M’emoli, G. Carlsson, Topological

Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition, Eurographics Symposium on Point-Based Graphics 2007.

◮ Examples and images from Tutorial of topological data

analysis part 3(Mapper algorithm): https://www.slideshare.net/Eniod/tutorial-of-topological- data-analysis-part-3mapper-algorithm

◮ Examples and images from Introduction to Topological Data

Analysis: https://www.slideshare.net/hendrikarisma/introduction-to- topological-data-analysis-59759836

◮ Examples and images from KeplerMapper:

https://mlwave.github.io/kepler-mapper/