So today for the CAA Seminar Series, we are having Professor Guillaume Calier from Université

Paris-Dauphine, and he will be speaking about Washington's Paris Center from a PDA perspective.

And of course we will have questions within or of course after the talk.

Yeah, so please, you can start.

Okay, so thank you very much for this nice introduction and for the presentation.

Of course, I would have been glad to visit Erlangen and to see Enrique in person and his group.

But, okay, I hope you're all doing well.

Seminars on Zoom are kind of a new experience to all of us.

So I'm going to start with a few pictures.

So my talk will be based on various joint works.

I'm going to talk about Wasserstein-Barre centers.

But before I enter the details, I'd like to show you some pictures you can find on the web

to explain what's going on, what is the problem.

So the Wasserstein-Barre Center in a nutshell is a way to interpolate between images or shapes

or something which are modeled by probability measures somewhere.

So I want to show you an example, which is this one.

So on top, you see there's a bunch of images which are very similar.

It's a bunch of ellipses.

And you would like to find a mean image, a sort of average of all these images.

And of course, you would like the mean, the average, to look like a pair of ellipses.

So on top is a sample of those images.

There are 30 of them, if I'm not mistaken.

And in the bottom, here is in figure A, it is what you obtain

if you just do the Euclidean average.

You do the 1 over n times the sum of these measures, if you want, these signals.

And of course, what you get is very bad because it's an image with support,

is a union of the support of the sample.

So it doesn't look at all like a pair of ellipses.

You can be slightly more clever and try to center the data.

This is a second figure, figure B.

It's slightly better, but still it looks like a bunch of spaghettis

and not like a pair of ellipses, right?

Now you can use more sophisticated devices that people use in data processing,

but it's still not, it doesn't look like a pair of ellipses.

And the last, the last interpolation scheme is based on what I'm going to describe

in a couple of minutes, which is this Barycenter, this Wasserstein Barycenter.

You see that there is some blurring, I will explain why,

but it still looks like a pair of ellipses, which is the average of all these samples.

So you can see a lot of other examples. This is what I found in the web.

You can also have different colors and try to average the colors as well.

You can change the topology, find interpolation between a crown and a cross

and a heart and two disjoint balls, something even more fancy between a cat and a spiral.

And you see that it is, at least from the visual point of view,

what the Wasserstein Barycenter does is exactly what you expect.

That the average shape exactly looks like what you want.

Let me show you a last example, which is this one, which is maybe a little bit more serious,

which is an interpolation between figures. So this is zero, one, a bunch of two, a bunch of three, and so on.

And this is what you obtain by interpolating them.

So it's a difficult thing to, when you have objects with a topology, like an eight,

if you have a bunch of eights, performing a mean which looks like an eight is quite a challenging problem.