INF556 — Topological Data Analysis

Coordinator : Steve Oudot (steve.oudot@inria.fr)

Link to the course's web page: click here

(Note: this course has changed its code: previously it was known as INF563)

Objectives :

Topological Data Analysis is an emerging trend in exploratory data analysis and data mining. It has known a growing interest and some notable successes in the recent years. The idea is to use topological tools to tackle challenging data sets, in particular data sets for which the observations lie on or close to non­trivial geometric structures that can fool classical techniques. Topological methods are indeed able to extract useful information about these geometric structures from the data, and to exploit that information to enhance the analysis pipeline. The objective of this course is to familiarize the students with this new topic lying at the confluence of pure mathematics, applied mathematics, and computer science. Emphasis is put on the methods and on their theoretical guarantees. Meanwhile, the lab sessions focus on challenging data sets, primarily multimedia data sets such as collections of images or 3d shapes.

Content :

The course is divided into nine lectures and nine exercise or lab sessions. These cover the main mathematical concepts and algorithmic tools involved in topological data analysis. The topics covered include:

• dimensionality reduction and its limitations,

• hierarchical versus density-based clustering,

• simplicial and singular homology,

• persistence theory,

• topological inference for data exploration,

• topological signatures for data classification,

• Reeb graphs and Mapper.

Suggested readings:

• Gunnar Carlsson. Topology and Data, Bulletin of the American Mathematical Society

• Herbert Edelsbrunner and John Harer, Computational Topoogy: An Introduction, AMS press

Language :

The course material is in English. Lectures can be taught either in French or in English, at the students' convenience.

Evaluation :

Graded lab session + final written exam.

Prerequisites :

• in mathematics: a fair knowledge of linear and bilinear algebra, plus some notions of general topology. A background in statistics (MAP433) or probability theory (MAP432) is desirable but not mandatory.
  

• in computer science: some knowledge of algorithms (INF421) and programming (INF431 or INF442 or modal INF474A).

Feedback from this year's experience: if you are from MP, then this course is for you; if you are not from MP, then this course is also for you, although you will have to work harder; in any case, the goals remain achievable and are worth the effort! Click here to see the students' feedback from 2015-2016 (about two thirds of the students answered the poll). You can visit the course's webpage for further information, or contact the coordinator if you have any questions or concerns.