CS 6170: Computational Topology

Topological Data Analysis for Data Scientists

Spring 2019

Schedule at a Glance (subject to change, see weekly schedule for details)

Mutually inclusive modules involving topological data analysis (TDA):
FP: TDA foundations and pipelines
ML: TDA, machine learning and statistics
DS: TDA in data science

SP: Special topics


Week
Date
Modules
Topic
Lecture
Book
Comments & Due Dates
1 1/8 Introduction Lecture 01 SP
1/10 FP Graphs, connected components Lecture 02 A.I
2 1/15 FP Surfaces, topological spaces, union-find Lecture 03 A.II
1/17 FP + DS Simplicial Complexes, sensor networks Lecture 04 A.III
3 1/22 FP + DS Various simplicial complexes, brain networks Lecture 05 A.III
1/24 FP Homology and computation Lecture 06 B.IV
4 1/29 FP TDA pipeline with persistence homology I Lecture 07 C.VII
1/31 FP TDA pipeline with persistence homology II Lecture 08 C.VIII
5 2/5 FP Manifolds, homology computation Lecture 09 SP Project 1 posted
2/7 FP TDA pipeline: computational tools Lecture 10 SP Final project team creation due.
6 2/12 ML TDA and machine learning: an introduction Lecture 11 SP
2/14 ML + DS Mapper and high-dimensional data I Lecture 12 SP
7 2/19 ML + DS Mapper and high-dimensional data II Lecture 13 SP
2/21 ML TDA, kernels, classification Lecture 14 SP Project 1 Due.
8 2/26 ML Kernel methods, learning and TDA I Lecture 15 SP Project 2 Posted.
2/28 ML Kernel methods, learning and TDA II Lecture 16 SP Final Project Proposal Requirement Posted
9 3/5 ML Kernel methods, learning and TDA III Lecture 17 C.VIII
3/7 ML Kernel methods, learning and TDA IV Lecture 18 C.VIII Final project proposal due.
10 3/12 Spring break!
3/14 Spring break!
11 3/19 FP Morse function Lecture 19 B.VI
3/21 FP Contour trees and Reeb graphs Lecture 20 B.VI Project 2 due (extended to 3/24).
12 3/26 ML + DS TDA and time series analysis Lecture 21 SP
3/28 ML + DS Elevation and protein docking I Lecture 22 SP
13 4/2 ML + DS Final Project Progress Presentation: Peer Review I Lecture 23 SP Final project progress report due.
4/4 FP Final Project Progress Presentation: Peer Review II Lecture 24 SP Project 3 (Bonus Project) Posted.
14 4/9 FP Elevation and protein docking II Lecture 25 SP
4/11* ML + DS Delaunay complexes, Alpha complexes, applications Lecture 26 SP
15 4/16 FP + DS Morse-Smale complexes Lecture 27 SP
4/18 ML + DS Cohomology, Future Directions and Discussions Lecture 28 SP Project 3 (Bonus Project) due.
16 4/23 Presentation Final project presentation (Tuesday, 9:10 a.m. - 10:30 a.m.)
17 4/29 (Monday) Presentation Final project presentation (Monday, 8:00 a.m. - 10:00 a.m.)
4/30 (Tuesday) Report Final project report due (Tuesday, 9:10 a.m.)

Weekly Schedule (subject to change)

Week 1

Lecture 01: Introduction, 1/8/2019
Course logistics, overview
Download Lecture 01

Lecture 02: FP 1/10/2019
Graphs and connected components
Download Lecture 02

Mandatory Reading and Tasks:
  1. [Carlsson2015]: Why Topological Data Analysis Works.
  2. [Carlsson2013]: Topological Data Analysis: A Framework for Machine Learning.
  3. [Carlsson2018]: Relationships, Geometry, and Artificial Intelligence.
  4. [Munch2017]: A User's Guide to Topological Data Analysis. DOI.
Recommended Reading:
  1. [Nielsen2018]: Neural Networks and Deep Learning. A good introduction to neural networks and deep learning.
  2. [EdelsbrunnerHarer2008]: Persistent Homology -- a Survey.
  3. [Carlsson2019]: Topology and Data.

Week 2

Lecture 03: FP, 1/15/2019
Surfaces, topological spaces, union-find
Download Lecture 03

Lecture 04: FP + DS, 1/17/2019
Simplicial Complexes, sensor networks
Download Lecture 04

Mandatory Reading and Tasks:
  1. [EdelsbrunnerHarer2010]: Textbook Chapters A.I and A.II.
  2. [DeSilvaGhrist2007]: Coverage in sensor networks via persistent homology.
Recommended Reading:
  1. [NicolauLevineCarlsson2011]: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. An example of TDA applied to biomedicine.
  2. [BotschSinghMemoli2007]: Topological methods for the analysis of high dimensional data sets and 3D object recognition. An introduction to the mapper algorithm used in [NicolauLevineCarlsson2011].

Week 3

Lecture 05: FP + DS, 1/22/2019
Various simplicial complexes, brain networks
Download Lecture 05

Lecture 06: FP, 1/24/2019
Homology and computation
Download Lecture 06

Mandatory Reading and Tasks:
  1. [EdelsbrunnerHarer2010]: Textbook Chapters A.III and B.IV.
  2. [DeSilvaGhrist2017a]: Homological Sensor Networks.
  3. [DeSilvaGhrist2007b]: Coverage in sensor networks via persistent homology.
  4. [Ripser]: a lean C++ code for the computation of Vietoris-Rips persistence barcodes. Install and play with Ripser to prepare for Project 1.
Recommended Reading:
  1. [Kun2015]: The Cech Complex and the Vietoris-Rips Complex.
  2. [WongPalandeWang2016]: Kernel Partial Least Squares Regression for Relating Functional Brain Network Topology to Clinical Measures of Behavior.

Week 4

Lecture 07: FP, 1/29/2019
TDA pipeline with persistence homology I
Download Lecture 07

Lecture 08: FP, 1/31/2019
TDA pipeline with persistence homology II
Download Lecture 08

Mandatory Reading and Tasks:
  1. [EdelsbrunnerHarer2010]: Textbook Chapters B.IV, C.VII and C.VIII.
  2. [Ripser]: a lean C++ code for the computation of Vietoris-Rips persistence barcodes. Install and play with Ripser to prepare for Project 1.
  3. [EdelsbrunnerHarer2008]: Persistent Homology -- a Survey.
Recommended Reading:
  1. [CarlssonZomorodianCollins2004]: Persistence Barcodes for Shapes.
  2. [Ghrist2008]: Barcodes: The Persistent Topology of Data.

Week 5

Lecture 07: FP, 2/05/2019
Project 1, homology computation, manifolds
Download Lecture 09
Download Project 01
Download Project 01 Data

Lecture 08: FP, 2/07/2019
Guest lecture by Professor Ingrid Hotz: topology in visualization - a practical perspective.
TDA pipeline: computational tools I
Download Lecture 10
Download Guest Lecture 10 by Prof. Ingrid Hotz.

Mandatory Reading and Tasks:
  1. [EdelsbrunnerHarer2010]: Textbook Chapters B.IV, C.VII and C.VIII.
  2. [HoferKwittNiethammer2017]: Deep Learning with Topological Signatures.
Recommended Reading:
  1. [CarlssonIshkhanovSilva2008]: On the Local Behavior of Spaces of Natural Images.
  2. [FasyKimLecci2015]: Introduction to the R package TDA.

Week 6

Lecture 11: ML 2/12/2019
TDA pipeline: computational tools. TDA and machine learning: an introduction.
Download Lecture 11

Lecture 12: ML + DS, 2/14/2019
Mapper and high-dimensional data I
Unsupervised learning, clustering, mapper algorithm.
Download Lecture 12

Mandatory Reading and Tasks:
  1. [HoferKwittNiethammer2017]: Deep Learning with Topological Signatures.
Recommended Reading:
  1. [FasyKimLecci2015]: Introduction to the R package TDA.

Week 7

Lecture 13: ML + DS 2/19/2019
Mapper and high-dimensional data II
Download Lecture 13

Lecture 14: ML, 2/21/2019
TDA, kernels, classification
Download Lecture 14

Mandatory Reading and Tasks:
  1. [KeplerMapper]: Get familiar with the KeplerMapper framework.
  2. [LumSinghLehman2013]: Extracting insights from the shape of complex data using topology.
Recommended Reading:
  1. [DeyMemoliWang2016]: Mutiscale Mapper: A Framework for Topological Summarization of Data and Maps.
  2. [DeyMemoliWang2017]: Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers.

Week 8

Lecture 15: ML, 2/26/2019
Kernel methods, learning and TDA I
Download Lecture 15
Download Project 02

Lecture 16: ML, 2/28/2019
Kernel methods, learning and TDA II
Download Lecture 16

Download Final Project Proposal Requirement

Week 9

Lecture 17: ML, 3/05/2019
Kernel methods, learning and TDA III
Download Lecture 17

Lecture 18: ML, 03/07/2019
Kernel methods, learning and TDA IV
Download Lecture 18

Mandatory Reading and Tasks:
  1. [AdamsChepushtanovaEmerson2016]: Persistence Images: A Stable Vector Representation of Persistent Homology.
Recommended Reading:
  1. [ChenNiBai2019]: A Topological Regularizer for Classifiers via Persistent Homology.
  2. [KerberMorozovNigmetov2016]: Geometry Helps to Compare Persistence Diagrams .

Week 10

Spring break!

Week 11

Lecture 19: ML + FP, 3/19/2019
More about Deep learning and TDA. Morse functions.
Download Lecture 19

Lecture 20: FP, 03/21/2019
Contour trees and Reeb graphs
Download Lecture 20

Mandatory Reading and Tasks:
  1. [EdelsbrunnerHarer2010]: Textbook Chapters B.VI.
Recommended Reading:
  1. [ChenNiBai2019]: A Topological Regularizer for Classifiers via Persistent Homology.

Week 12

Lecture 21: ML + DS, 3/26/2019
TDA and time series analysis.
Download Lecture 21

Lecture 22: FP, 03/28/2019
Elevation and protein docking I
Download Lecture 22

Final Project Progress Report Formatting Requirement

Mandatory Reading and Tasks:
  1. [EdelsbrunnerHarer2010]: Textbook Chapters B.VI., B.VII
Recommended Reading:
  1. [ChoudhuryWangRosen2012]: Topological Analysis and Visualization of Cyclical Behavior in Memory Reference Traces.
  2. [AgarwalEdelsbrunnerHarer2006]: Extreme Elevation on a 2-Manifold.
  3. [WangAgarwalBrown2005]:Coarse and reliable geometric alignment for protein docking.

Week 13

Lecture 23, 4/2/2019
Final Project Progress Presentation: Peer Review I.

Lecture 24, 4/4/2019
Final Project Progress Presentation: Peer Review II.
Download Project 03 (Bonus Project)

Mandatory Reading and Tasks:
  1. Remember to make progress towards your final project!
Recommended Tasks:
  1. [KeplerMapper]: Get familiar with the KeplerMapper framework.

Week 14

Lecture 25, 4/9/2019
Elevation and protein docking II.
Continuation of Lecture 22.

Lecture 26, 4/11/2019
Guest Lecture by Prof. Jeff Phillips: Delaunay complexes, Alpha complexes, applications

Recommended Readings:
  1. [CarlssonZomorodian2009]: The Theory of Multidimensional Persistence.
  2. [CorbetFugacciKerber2018]: A Kernel for Multi-Parameter Persistent Homology.
  3. [RIVET]: Program for the visualization and analysis of two-parameter persistent homology.

Week 15

Lecture 27, 4/16/2019
Morse-Smale complexes.
Download Lecture 27

Lecture 28, 4/11/2019
Cohomology, Future Directions and Discussions
Download Lecture 28

Recommended Readings:
  1. [GerberRubelBremer2012]: Morse-smale regression.
  2. [GerberBremerPascucci2010]: Visual exploration of high dimensional scalar functions.

Week 16

Presentation, 4/23/2019 (Tuesday) 9:10 a.m. - 10:30 a.m.
Final Projects Presentation I.

Week 17

Presentation, 4/29/2019 (Monday) 8:00 a.m. - 10:00 a.m.
Final Projects Presentation II.

Final Project Report Formatting Requirement

Final Project Report Due on 4/30/2019 (Tuesday) 9:10 a.m.

Recommended Readings for Course Projects (TDA + X)

ML

  1. [Phillips2018]: Mathematical Foundations for Data Analysis. A good introduction to data mining including some machine learning algorithms.
  2. [Nielsen2018]: Neural Networks and Deep Learning. A good introduction to neural networks and deep learning.
  3. [Daumé]: A Course in Machine Learning, by Hal Daumé III.
  4. [Shalev-Shwartz Ben-David]: Understanding Machine Learning: From Theory to Algorithms, by Shai Shalev-Shwartz and Shai Ben-David.
  5. [GoodfellowBengioCourville2016]: Deep learning, MIT press, 2016, by Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio.
  6. [VivekSrikumar]: CS 5350/6350 Machine Learning, University of Utah. Lecture notes by Vivek Srikumar.

TDA + ML

  1. [PunXiaLee2018]: Persistent-Homology-based Machine Learning and its Applications - A Survey.
  2. [ChenQuadrianto2016]: Clustering High Dimensional Categorical Data via Topographical Features.

TDA + Materials Science

  1. [MSRI2018]: Mathematical Sciences Research Institute (MSRI) Workshop, Hot Topics: Shape and Structure of Materials, Oct. 2018. See videos and slides of invited talks.

TDA + Astronomy

  1. [CMUTopStat2018]: See AstroStatistics.

TDA + Imaging + Computer Vision

  1. [WuChenWang2017]: Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration.
  2. [ChenMetaxasWang2017]: Cardiac Trabeculae Segmentation, an Application of Computational Topology.
  3. [ChenFreedmanLampert2011]: Enforcing topological constraints in random field image segmentation.

Resources

TDA Classes on the Web:

CPS296.1: Computational Topology By Herbert Edelsbrunner @ Duke
CSE 5339: Computational Topology and Data Analysis By Tamal K Dey @ OSU
CS 598: Computational Topology By Jeff Erickson @ UIUC
CS 468: Introduction to Computational Topology By Afra Zomorodian @Stanford
CSCI 491/591: Computational (Geometry and) Topology By Brittany Terese Fasy @ MSU

Other TDA Resources on the Web:

AppliedTopology.org

Topology Classes/Books on the Web:

Topology Illustrated by Peter Saveliev @ Marshall

Data on the Web:

Data.gov
Data.un.org
Awesome Public Datasets
Stanford Large Network Dataset Collection (SNAP)
Network Repository

Coding resources on the Web:

Interactive Data Visualization for the Web, 2nd Edition by Scott Murray (2017)