Okay, thanks a lot, Martin.

It's a great pleasure to be here.

It's a new experience for me.

It's the first time I do this.

I hope everything will go well, I'm sure.

So I wanted to do some kind of overview of optimal transport from a computational perspective.

And also highlight some recent work that we have been doing, which are about trying to

scale optimal transport to high dimension, both in terms of competition time and also

in terms of statistical performance.

It's a joint work with a lot of people, in particular, of course, with Marco Cuturi,

who is my main collaborator on these lines of ideas.

And it's also in large part the PhD work of Haute-Jean Ve, at least for the machine learning

part.

All right, before starting, just some shameless advertisement for this book.

So feel free to go online to retrieve the PDF.

It will detail everything that I would say today and much more.

And it's totally free and open source.

And you also have Python codes to go with it.

All right, so first a bit of motivation.

Why optimal transport?

First, actually why trying to model problems with probability distribution, histograms,

and so on.

I think it's fairly obvious that in the most imaging processing field, there is some opportunity

for using this type of models.

You can think about modeling the colors or modeling feature of image, also for shapes

or neuroscience.

I think it really makes sense to try to manipulate some kind of a density of points or density

of electric activity, for instance, in the brain.

And I would say more recently in a high dimensional setup, both supervised learning and unsupervised

learning, there is an interest in trying to introduce some tools to manipulate densities.

For instance, in natural language processing, there is a breakthrough the last few years,

I would say, that tries to model the data sets of text using world embedding.

So you see the text as a point cloud in a very high dimensional space for doing supervised

learning, for instance.

And also for, for instance, generative models of images.

There is this idea also trying to manipulate the image, I think, as a big point cloud in

very high dimensional space.

Okay, so in all these type of problems, the key question you're asking is to try to fit

some prior models, some parametric model, let's call it alpha, depending on from parameter

theta to a huge point counts.

So it's a basic problem in image processing, shape matching, generative models, where you

try to model the data using a deep nets.

And really the goal is to try to fit the model, to try to make the model as close as possible

to the point cloud.

So you can frame this question as trying to minimize some kind of discrepancy capital

D between your model and your data.

It's pretty obvious.

And I would say many problems can be framed like this.

And the key question here is what should be the capital D that you want to use to address

this efficiently.