Where to start learning about topological data analysis?

I was wondering if anyone could help me out with finding a nice introductory introductory text for topological data analysis (I'm speaking as somebody who has two semesters of experience with topology, and much less experience with data analysis). Are there any self-contained elementary resources on the subject? And if not, is there a sort of road map for the subject (i.e. a generally agreed upon sequence of topics that I should study)?

I saw a nice overview here: http://www.cs.dartmouth.edu/~afra/papers/ams12/tda.pdf, and that piqued my interest in the topic. Thanks in advance for the help!

• I know nothing about data analysis, but, in addition to general topology that you probably took, you would need some basic algebraic topology as well (simplicial complexes, homology, cohomology, Poincare duality). – Moishe Kohan May 29 '14 at 15:38
• mathoverflow.net/questions/141157/… on MathOverflow may also be of interest. – J W Oct 16 '14 at 17:54
• That link seems to be obsolete. – Michael Hardy Dec 20 '16 at 6:28
• David Austin, Finding Holes in the Data, Feature Column of the AMS, December 2016. – Rodrigo de Azevedo Dec 20 '16 at 7:37

At the time of writing, I'm not aware of any books that are very specifically about topological data analysis (TDA), apart from the collected papers in Topological Methods in Data Analysis and Visualization and its two sequels, but there are a handful on computational topology that contain valuable background and details for TDA. Gurjeet has already mentioned Afra Zomorodian's Topology for Computing. Others include:

At the moment, knowledge of statistics does not appear to be a prerequisite, although there is some interesting work in that direction at CMU: http://www.stat.cmu.edu/topstat/. It is helpful to be comfortable with multivariable calculus, linear algebra, introductory abstract algebra (especially group theory) and basic point-set topology. Prior acquaintance with algebraic topology and manifolds would be even better. For comparison purposes, it may be interesting to look into clustering algorithms such as $$k$$-means and hierarchical clustering.

You may want to take a look at Peter Saveliev's Topology Illustrated (which is indeed liberally and helpfully illustrated, so the title is accurate) with its emphasis on homology, and Robert Ghrist's Elementary Applied Topology for a broad-ranging invitation to applied topology. Michael Robinson's Topological Signal Processing could also be of interest.

Update: The importance of the whole: topological data analysis for the network neuroscientist by Sizemore, Phillips-Cremins, Ghrist & Bassett is a nice introductory paper for a first look at TDA.

• As an update to this post, Saveliev's text is no longer a draft and has published a first edition as of 2016. – Chill2Macht Jun 12 '17 at 10:09
• @Chill2Macht: Thank you! I have updated the link. – J W Jan 1 at 18:01

Afra's paper is a really good introduction. He has also written a really accessible book which is accessible here:

http://www.amazon.com/Computing-Cambridge-Monographs-Computational-Mathematics/dp/0521136091/ref=sr_1_1?ie=UTF8&qid=1401393274&sr=8-1&keywords=afra+zomorodian

I would recommend playing with some software. Here's some:

http://comptop.stanford.edu/programs/

• The linked book seems to be about something else, at least if you believe the description. It seems to be about computational aspects of topology, as opposed to using topological methods for data analysis a la Gunnar Carlsson. – Cheerful Parsnip May 29 '14 at 20:03
• The link to CompTop at Stanford doesn't seem to work. Is it temporarily down or have the resources been moved elsewhere? – J W Oct 16 '14 at 17:36
• @GrumpyParsnip: comparing the table of contents of Topology for Computing with Zomorodian's article Topological Data Analysis shows that there is significant overlap. Also, Zomorodian has written papers together with Carlsson on persistent homology. – J W Oct 17 '14 at 5:25
• @JW: you're right, I was too hasty. There is some discussion on TDA in this book. Your linked paper also looks like a good reference. – Cheerful Parsnip Oct 17 '14 at 14:41
• @GrumpyParsnip: no problem. By the way, the linked paper is simply from the original question. – J W Oct 17 '14 at 14:56