Topological Data Analysis (TDA) is an emerging trend in exploratory data analysis and data mining. It has known a growing interest and some notable successes (such as the identification of a new type of breast cancer, or the classification of NBA players) in the recent years. Indeed, with the explosion in the amount and variety of available data, identifying, extracting and exploiting their underlying structure has become a problem of fundamental importance. Many such data come in the form of point clouds, sitting in potentially high-dimensional spaces, yet concentrated around low-dimensional geometric structures that need to be uncovered. The non-trivial topology of these structures is challenging for classical exploration techniques such as dimensionality reduction. The goal is therefore to develop novel methods that can reliably capture geometric or topological information (connectivity, loops, holes, curvature, etc) from the data without the need for an explicit mapping to lower-dimensional space. The objective of this course is to familiarize the students with these new methods, lying at the interface between pure mathematics, applied mathematics, and computer science.

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- lectures: Wednesdays 8:30 - 10:30, amphi Monge
- lab/exercise sessions: Wednesdays 10:45 - 12:45, amphi Monge

Important: bring pens and paper to take notes during the lectures, and your laptop for the lab sessions.

**Before you come** to the first lab session, you are advised to:

- Install a Java compiler like the Java Development Kit, and a C++ compiler like gcc. You can use whatever IDE you prefer, e.g. Eclipse or Code::Blocks.
- Make sure your laptop is equipped with R as well. See the
*Set Up*section on the first TD page for more details on how to install the software and configure your environment. - Get familiar with the R language a bit. For this I recommend the following tutorial, you can restrict yourself to the "R introduction" section (introductory page + pages on basic data types).

**Course evaluation:**

- Midterm (TD noté): 50%
- Final written exam (3 hours): 50%

- a brief introduction to the course
- the complete syllabus of the course

- H. Edelsbrunner, J. Harer.
*Computational Topology: An Introduction*. AMS Press, 2009. A good introduction to applied topology, including TDA. Well-suited for this course. This book is not available at the library, however it was compiled from the following course notes, which you can download instead. - S. Oudot.
*Persistence Theory: From Quiver Representations to Data Analysis*. AMS Surveys and Monographs, Vol. 209, 2015. A comprehensive treatment of persistence theory, perhaps too advanced for this course, but in principle you should be able to read it by the end of the course! Five printed copies are available at the library. - James R. Munkres. Elements of Algebraic Topology. Perseus, 1984. A general introduction to algebraic topology, which you can consult (especially its first chapter) for more background on homological algebra. One printed copy is available at the library.
- Trevor Hastie, Robert Tibshirani and Jerome Friedman. The Elements of Statistical Learning (2nd edition). Springer-Verlag, 2009. An excellent reference in learning. Chapter 14 covers the material on dimensionality reduction addressed during the first session. You can download a pdf version of the book here.

Notes, slides | Session 1: Dimensionality Reduction | TD 1 | Jan. 04 2017 |

Slides intro clustering, Slides mode-seeking Notes mode-seeking, Notes degree-0 persistence ToMATo's webpage |
Session 2: Clustering | TD 2a, TD 2b | Jan. 11 2017 |

Notes homology, book homology | Session 3: Homology I | PC 3-4, solution | Jan. 18 2017 |

Session 4: Homology II | Jan. 25 2017 | ||

Notes persistence,
Slides persistence
book persistence 1, book persistence 2 |
Session 5: Persistence I | TD 5 (noté, due on or before Jan. 31) | Fev. 01 2017 |

Notes inference, Slides inference | Session 6: Persistence II / Topological Inference | TD 6 | Fev. 08 2017 |

Notes signatures, Slides signatures
Notes kernels+stats, Slides kernels+stats |
Session 7: Topological Signatures | TD 7-8 | Fev. 22 2017 |

Session 8: Kernels and Statistics on Signatures | Mar. 01 2017 | ||

Notes, Slides | Session 9: Reeb graph and Mapper | PC 9 (révisions) | Mar. 08 2017 |

Feel free to come and ask us directly if you are looking for an internship in TDA.

The subject has known a steady development throughout the last decade or so. Here is a snapshot of the current TDA community:

Last update: Dec. 31 2016.