Large Deviation, Optimal Transport and Machine Learning Reference

Question

I am looking for references (books/sites/articles) on the following three subjects: Large Deviation, Optimal Transport and Machine Learning References. I would like works which involve any of them individually, or jointly. My motivation for this is the following course description which used to be taught at my school:

Two mathematical frameworks: optimal transport and large deviation theory, can be combined to provide tools and methods of analysis for some central tasks in data science: classification, clustering and regression, dimensionality reduction, conditional probability estimation, prediction of rare events. Recurring themes will include the power of the composition of simple functions to capture complex behavior - as demonstrated by neural networks - and of the characterization of both the available observations of the system under study and the parameters tuned to model it in terms of underlying probability distributions.

If you know of any core books on either Large Deviation or Optimal Trasport that can be grasped at a beginning graduate level, I would much appreciate it!

thank you!

Answer 1

For optimal transport, have a look at this book: "https://arxiv.org/abs/1803.00567". For large deviations, the standard Springer reference book by Dembo and Zeitouni is a bit tough; the short book by van Hollander is somewhat easier.

I will say, though, that "beginning graduate level" is possibly a stretch. The recent optimal transport book by Santambrogio (also good) says point blank that you'll have a tough time if you don't already have a decent background in functional analysis and measure theory, and I think the same is true for large deviations.

Answer 2

For large deviation theory in general (and applications in statistical mechanics), see R. Ellis' notes: https://people.math.umass.edu/~rsellis/pdf-files/Les-Houches-lectures.pdf

Answer 3

Just in order to add also one particular and comparably famous reference combining aspects of OT and large deviations:

S Adams, Nicolas Dirr, M Peletier, J Zimmer; From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage

https://researchportal.bath.ac.uk/en/publications/from-a-large-deviations-principle-to-the-wasserstein-gradient-flo

Answer 4

Probably a bit late, but I can supplement the other answers.

For general introduction to the topics you mentioned :

Optimal transport : Villani's "Optimal Transport Old and New" book is great.

Large Deviations : I think Dembo and Zeitouni's book is good. Hollanders book is ok, but contains a bit less content.

Machine Learning : There is a lecture course on youtube which is good https://www.youtube.com/watch?v=_XmGyd4smUs&list=PL8FnQMH2k7jzhtVYbKmvrMyXDYMmgjj_n&index=2 .

There has been a lot of work done w.r.t Large Deviations (Sanov theorem, and Friedlin-Wentzell theory) in determining the cost function of gradient flows in measure spaces (this topic branches probability theory, thermodynamics, differential geometry, optimal transport) . See Peletiers works, in particular https://arxiv.org/abs/1312.7591.