Okay, so thank you very much for the invitation and also like you welcome everyone to One

World Seminar. So it's a good way not to travel too much. Okay, so there's a

representer theorem in the title. You all know what's machine learning and inverse

problems and so representer theorem really refers to a way of actually

representing the solution of sort of optimization problems. Okay, so now to

set the stage we have like the variational formulation of inverse

problems which is sort of the standard way of formulating this kind of problem

mathematically. So you have some unknown objects that since we're an integral

operator, a microscope, an MRI, you have sources of noise, this produces blurring

and then the problem is given that those measurements which are usually

discrete, how can we recover here the unknown here in concentration of your

force in 3D. And so the usual way then is to state that as an optimization

problem and what you are doing you are sort of enforcing a consistency between

let's say your reconstruction here, simulated measurements of your

reconstruction compared to your measurements wide and of course because

the problem is usually ill-posed, you're also like imposing some regularization

like for example putting a penalty on the L1 or L2 norm of some operator

applied to your signal. Okay, so that's inverse problems but in fact learning,

there was machine learning in the title, is also a linear inverse problem but an

infinite dimensional one. And so here the situation is similar so you're

getting given like a series of data points so x are some data points and

maybe y should be some outcome that you associate to a given pattern here. Now

you want to find a function that goes from Rn to R, such that the function

applied to your data point is your predictor here ym. And again this is a

ill-posed problem in principle and so early on people have reformulated that

using regularization theory and so again here you are introducing energy

functional for example L2 norm of some operator like the Laplacian applied to

your function f and you're trying here to solve then this data consistency

constraint here that you want your y to be, I mean the f of x here at the data

points be relatively close to your y and then here you are minimizing here

this regularization term here. And this you can also reframe in terms of

linear least squares like least square fit where using Lagrange multipliers and

have this kind of problem. And now if you solve that kind of problem remarkably

using the theory of reproducing kernel Hilbert spaces you can find that the

solution is a kernel estimator and this is very much the foundation of all

classical machine learning. And here is this famous representer theorem for

machine learning and what does it tell us? It tells us actually that the

minimizer of that over a certain Hilbert space H and now I remind you this is an

infinite dimensional problem because f is just a function in a certain Hilbert

space and so the representer theorem tells us that the solution of that

problem rather remarkably is a linear combination of kernels. Okay so those

kernels are those guys are H that depends here on x which is the free

variable and xm here is the location of your data points. And now what is this

RH? This is the so-called reproducing kernel of a reproducing kernel Hilbert

space and such spaces are actually characterized in the sense that they

exist a single kernel with something that goes from RD to RD into R so X being

in RD and so it is a it is a kernel that if you fix one of the variable this

thing lies in the Hilbert space and the other thing is the reproducing property

that if you now fix at that particular location and leave this variable open

and make the inner product with a function f that this will sample the