Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Tutorial of
Topological Data Analysis
Tran Quoc Hoan
@k09hthaduonght.wordpress.com/
Paper Alert 2016-04-15, Hasegawa lab.,...
My TDA = Topology Data Analysis ’s road
TDA Road 2
Part I - Basic concepts &
applications
Part II - Advanced TDA
computati...
TDA Road Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
Mapper Algorithm
Basic motivation
Mapper Algorithm 4
Basic idea
Perform clustering at different “scales”, track how
clusters change as scale...
Morse theory
Mapper Algorithm 5
Basic idea
Describe topology of a smooth manifold M using level
sets of a suitable functio...
Reeb graphs
Mapper Algorithm 6
• For each t in R, contract each
component of f-1(t) to a point
• Resulting structure is a ...
Mapper
Mapper Algorithm 7
The mapper algorithm is a generalization of this procedure (Singh-
Memoli-Carlsson)
Input
✤ Filt...
Workflow - Illustration
Mapper Algorithm 8Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
f cou...
Workflow - Illustration
Mapper Algorithm 9Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
f cou...
Workflow - Illustration
Mapper Algorithm 10Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
f co...
Mapper in practice
Mapper Algorithm 11
Input
✤ Filter (continuous) function f: P -> R
✤ Cover L of im(f) by open intervals...
Mapper in practice
Mapper Algorithm 12Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
Mapper in practice
Mapper Algorithm 13Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
Mapper in practice
Mapper Algorithm 14Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
Mapper in practice
Mapper Algorithm 15
Parameters
✤ Filter (continuous) function f: P -> R
✤ Cover L of im(f) by open inte...
Filter functions
Mapper Algorithm 16
Choice of filter function is essential
• Some kind of density measure
• A score measur...
Filter functions
Mapper Algorithm 17
Eccentricity
Density
- How close the point lies to the “center” of the point cloud.
-...
Mapper in applications
Mapper Algorithm 18
Extracting insights from the shape of complex data using topology,
Lum et al., ...
Mapper in practice
Mapper Algorithm 19
1. Clustering
2. Feature selection
Mapper in clustering
Mapper Algorithm 20
(1) Compute the Mapper
(2) Detect interesting topological substructures
(“loops”,...
Mapper Algorithm 21
Extracting insights from the shape of complex data using topology,
Lum et al., Nature, 2013
f: 1st and...
Mapper Algorithm 22
Extracting insights from the shape of complex data using topology,
Lum et al., Nature, 2013
Detect new...
Mapper Algorithm 23
Innate and adaptive T cells in asthmatic patients: Relationship
to severity and disease mechanisms, Hi...
Mapper in feature selection
Mapper Algorithm 24
(1) Compute the Mapper
(2) Detect interesting topological substructures
(“...
Mapper Algorithm 25
Extracting insights from the shape of complex data using topology,
Lum et al., Nature, 2013
Goal: dete...
Mapper Algorithm 26
Topological Data Analysis for Discovery in Preclinical Spinal Cord
Injury and Traumatic Brain Injury, ...
Select Parameters
Mapper Algorithm 27
parameter r
parameter g
parameter δ
parameter f
• Small r -> fine cover 

(close to R...
Select Parameters
Mapper Algorithm 28
Example: P in R2 sampled from known distribution
f = density estimator, r = 1/30, g ...
Reference links
Mapper Algorithm 29
• INF563 Topological Data Analysis Course

http://www.enseignement.polytechnique.fr/in...
Upcoming SlideShare
Loading in …5
×

Tutorial of topological data analysis part 3(Mapper algorithm)

3,329 views

Published on

Mapper algorithm

Published in: Technology

Tutorial of topological data analysis part 3(Mapper algorithm)

  1. 1. Tutorial of Topological Data Analysis Tran Quoc Hoan @k09hthaduonght.wordpress.com/ Paper Alert 2016-04-15, Hasegawa lab., Tokyo The University of Tokyo Part III - Mapper Algorithm
  2. 2. My TDA = Topology Data Analysis ’s road TDA Road 2 Part I - Basic concepts & applications Part II - Advanced TDA computation Part III - Mapper Algorithm Part V - Applications in… Part VI - Applications in… Part IV - Software Roadmap He is following me
  3. 3. TDA Road Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/ Mapper Algorithm
  4. 4. Basic motivation Mapper Algorithm 4 Basic idea Perform clustering at different “scales”, track how clusters change as scale varies Motivation • Coarser than manifold learning, but still works in nonlinear situation • Extract meaningful geometric information about dataset • Efficiently computable (for large dataset) Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition. G Singh, F Mémoli, GE Carlsson - SPBG, 2007
  5. 5. Morse theory Mapper Algorithm 5 Basic idea Describe topology of a smooth manifold M using level sets of a suitable function h : M -> R • Recover M by looking at h-1((∞, t]), as t scans over the range of h • Topology of M changes at critical points of h
  6. 6. Reeb graphs Mapper Algorithm 6 • For each t in R, contract each component of f-1(t) to a point • Resulting structure is a graph
  7. 7. Mapper Mapper Algorithm 7 The mapper algorithm is a generalization of this procedure (Singh- Memoli-Carlsson) Input ✤ Filter (continuous) function f: X -> R ✤ Cover L of im(f) by open intervals: Method ✤ Cluster each inverse image f-1(Lα) into various connected components ✤ The Mapper is the nerve of V • Clusters are vertices • 1 k-simplex per (k+1)-fold intersection connected cover V ✤ Color vertices according to average value of f in the cluster k i=0Vi 6= ;, V0, ..., Vk 2 V
  8. 8. Workflow - Illustration Mapper Algorithm 8Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/ f could be in n-dimension
  9. 9. Workflow - Illustration Mapper Algorithm 9Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/ f could be in n-dimension
  10. 10. Workflow - Illustration Mapper Algorithm 10Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/ f could be in n-dimension
  11. 11. Mapper in practice Mapper Algorithm 11 Input ✤ Filter (continuous) function f: P -> R ✤ Cover L of im(f) by open intervals: Method ✤ Cluster each inverse image f-1(Lα) into various connected components in G ✤ The Mapper is the nerve of V connected cover V ✤ Color vertices according to average value of f in the cluster - Point cloud P with metric dP - Compute neighborhood graph G = (P, E) • Clusters are vertices • 1 k-simplex per (k+1)-fold intersection k i=0Vi 6= ;, V0, ..., Vk 2 V (intersections materialized by data points)
  12. 12. Mapper in practice Mapper Algorithm 12Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
  13. 13. Mapper in practice Mapper Algorithm 13Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
  14. 14. Mapper in practice Mapper Algorithm 14Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
  15. 15. Mapper in practice Mapper Algorithm 15 Parameters ✤ Filter (continuous) function f: P -> R ✤ Cover L of im(f) by open intervals: ✤ Neighborhood size δ Example: uniform cover L • Resolution / granularity: r (diameter of intervals) • Gain: g (percentage of overlap) range scale geometric scale
  16. 16. Filter functions Mapper Algorithm 16 Choice of filter function is essential • Some kind of density measure • A score measure difference (distance) from some baseline • An eccentricity measure Statistics
 Mean/Max/Min Variance n-Moment Density … Machine Learning
 PCA/SVD Auto encoders Isomap/MDS/TSNE SVM Distance Error/Debugging Info … Geometry
 Centrality
 Curvature
 Harmonic Cycles …
  17. 17. Filter functions Mapper Algorithm 17 Eccentricity Density - How close the point lies to the “center” of the point cloud. - How close the point to the surrounding points
  18. 18. Mapper in applications Mapper Algorithm 18 Extracting insights from the shape of complex data using topology, Lum et al., Nature, 2013 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 Topological Methods for Exploring Low-density States in Biomolecular Folding Pathways, Yao et al., J. Chemical Physics, 2009 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 Innate and adaptive T cells in asthmatic patients: Relationship to severity and disease mechanisms, Hinks et al., J. Allergy Clinical Immunology, 2015 ✤ ✤ ✤ ✤ ✤ ✤
  19. 19. Mapper in practice Mapper Algorithm 19 1. Clustering 2. Feature selection
  20. 20. Mapper in clustering Mapper Algorithm 20 (1) Compute the Mapper (2) Detect interesting topological substructures (“loops”, “flares”) (3) Use substructure to cluster data select parameters Not easy (Tutorial part 1 + 2)
  21. 21. Mapper Algorithm 21 Extracting insights from the shape of complex data using topology, Lum et al., Nature, 2013 f: 1st and 2nd SVD r = 120, g = 22% PCA can show the Republican/ Democrat cluster but TDA gives more information House Party representative grouping Point: member of the House PCA
  22. 22. Mapper Algorithm 22 Extracting insights from the shape of complex data using topology, Lum et al., Nature, 2013 Detect new clusters for NBA players
  23. 23. Mapper Algorithm 23 Innate and adaptive T cells in asthmatic patients: Relationship to severity and disease mechanisms, Hinks et al., J. Allergy Clinical Immunology, 2015 The TDA used 62 subjects with most complete data. f: 1st and 2nd SVD r = 120, g = 14%, equalized
  24. 24. Mapper in feature selection Mapper Algorithm 24 (1) Compute the Mapper (2) Detect interesting topological substructures (“loops”, “flares”) (3) Select features that best discriminate data in substructure select parameters Kolmogorov-Smirnov test on (substructure) feature vs. (whole dataset) feature, select features with low p-val
  25. 25. Mapper Algorithm 25 Extracting insights from the shape of complex data using topology, Lum et al., Nature, 2013 Goal: detect factors that influence survival after therapy in breast cancer patients Points: breast cancer patients that went through specific therapy PCA/Single-linkage clustering cannot see this f: eccentricity r = 1/30, g = 33%
  26. 26. Mapper Algorithm 26 Topological Data Analysis for Discovery in Preclinical Spinal Cord Injury and Traumatic Brain Injury, Nielson et al., Nature, 2015
  27. 27. Select Parameters Mapper Algorithm 27 parameter r parameter g parameter δ parameter f • Small r -> fine cover 
 (close to Reeb) (sensitive to δ) • Large r -> rough cover 
 (less sensitive to δ) • g ≈ 1 -> more points inside intersections , less sensitive to δ but far from Reeb • g ≈ 0 -> controlled Mapper dimension, close to Reeb • Large δ -> fewer nodes, clean Mapper but far from Reeb (more straight lines) • Small δ -> distinct topological structure but lots of nodes (noisy) • Depend mostly on the dataset coordinate, density estimation, eccentricity, eigenvector
  28. 28. Select Parameters Mapper Algorithm 28 Example: P in R2 sampled from known distribution f = density estimator, r = 1/30, g = 20% δ = percentage of the diameter of X Image source: http://www.enseignement.polytechnique.fr/informatique/INF563/
  29. 29. Reference links Mapper Algorithm 29 • INF563 Topological Data Analysis Course
 http://www.enseignement.polytechnique.fr/informatique/INF563/ • AYASDI
 http://www.ayasdi.com/ • …

×