All right, thanks everybody for organizing this.

It's a pleasure.

It's the first time I do this.

So I'm sharing my iPad.

If there is any connection problem, let me know.

I changed the title with respect to the one I proposed a couple of weeks ago.

I tried to make a kind of a broad overview of work that has been going on in the last few years

with a number of collaborators.

Daniele Raffaello, Luigi Giacomo from my group in Genova,

especially Alessandro Rudy from Paris and others,

Barath, Mik, Alessandro and Mikkel.

So the overall theme is this idea that one of the great challenges in current machine learning,

it works really well, but it's very demanding from a computational point of view.

So we always have to struggle finding computational resources.

And the idea of trying to find ways to keep the good performances while

provably reducing the computational cost is largely a challenge.

And it's perhaps not the top priority of industries that have large computational facilities.

So the attempt here is to somewhat investigate ideas in the direction

of keeping good performances while reducing computational costs.

So as I said, I kind of give a long overview.

And I'm going to be switching among different topics.

And depending on time, I might skip some of this.

So the first thing is to introduce the main idea.

It is what I call the Nystrom random projection.

So it's a form of random projection, which is data-driven.

And here, what I'm going to do first is going to introduce them

and then provide different views.

On the one hand, this might confuse some.

Hopefully, these will also be a way for each one of you,

depending on the different backgrounds,

to somewhat find your favorite way to look at the problem.

So the basic idea is that we start

from the basic observation that we call x hat.

The right, I see stuff happening now.

OK, so we call x hat the data matrix.

And we think of a data matrix where

n is the number of data points and d

is the dimension of the data.

So they come in natural, vectorial form.

And we think of both n and d be extremely large.

And the big observation is that oftentimes,

dealing with such big matrices, it's extremely cumbersome,

especially for memory constraint.

Because in some sense, handling these matrix,

storing this matrix becomes a problem.

So a very simple idea, the idea I start from,

is that you have reduced dimensionality.

So the idea is that you're going to hit your matrix

at the right with the matrix that I call here s.

This matrix has the same dimension