Yes, I'm going to talk today about a very generic title almost, but this is sort of

work that I've been, well, amongst other work I've been recently been working on, on the

intersection of machine learning and inverse problems in some ways.

And yeah, as Leon said, it will actually involve Breckmann distances, which I believe is one

of the few research groups where I can probably talk about Breckmann distances without even

having to explain them.

Although maybe this has changed, maybe it's not the case anymore, who knows.

We'll see.

Okay, let's get started.

What I should mention first is that this is predominantly work of my PhD student, Victor,

who you see here, who's actually not at Queen Mary, he's at Cambridge.

So when I was still at Cambridge, he became my PhD student and therefore, yeah, he's still

my PhD student.

And this is predominantly his work.

Okay.

And what am I going to talk today about in terms of the outline, in terms of details?

So what I first want to say, so Leon instructed me that I should try to give a talk more like

on a master level, on a basic level for a mixed audience.

So I will first talk about perceptrons and what perceptrons are.

It may very well be that you all very well know what perceptrons are, but it's probably

always good to also set the notation and to recall the basic properties and concepts.

And then we come to the training or the learning of perceptrons and how we've got to paraphrase

a very old classical algorithm, namely the Rosenblatt algorithm, and see how we can actually

formulate it as a proper gradient-based algorithm for which we will define a Breckmann loss

or a loss based on Breckmann distances.

And then if time permits, actually this might be a big if, I even will talk a little bit

about the extension to general artificial neural networks, which is current work in

progress.

So we don't have a paper on this yet.

We have a small paper on the first part.

And I will then conclude with some conclusions and outlook.

Okay.

Let's dive straight in into the concept of perceptrons.

So yeah, back in the fifties and sixties of the previous century, the idea of perceptrons

were born and there were basically simplistic models of biological neurons.

So the idea was to take a biological neuron and to create an artificial neuron in a simplistic

way where basic parts of the perceptron, such as the dendrites that take the inputs, the

cell body and the nucleus that process these inputs that the dendrites receive and the

sort of axons which transmit a potential output of a neuron that they are being modeled in

form of an artificial neuron.

And the most popular artificial neuron, originally proposed by Frank Rosenblatt, is the perceptron

or used to be the perceptron.

And what does the perceptron do?

So a perceptron takes a couple of inputs, which traditionally are binary.

Today they do not necessarily have to be binary anymore, which here in this graphical case,

and that actually reminds me that I do have a pointer and I should obviously also use

the pointer so that we have these inputs here, x1, x2, x3, for example.

So here we have a perceptron with three inputs and one output.

The idea is that the perceptron takes these inputs in the same way that a biological neuron

would process inputs and then that the cell body is being or that the process of whether