We are now in the chapter about unsupervised learning.

And up to now, we did do some remarks on linear algebra.

Nevertheless, I always find it astonishing how many effort

you have to do in this equation stuff, which later I

can explain you much easier by using architectures.

So see this as an exercise that the thinking

and architectures is a valuable style of thinking.

So we have been here.

Yeah.

So let's assume we have a set of points in the area here,

in the two-dimensional area here.

And we would like to look for a unit vector v, which

is going in the direction of most of the data points here.

So I speak about the blue vector here.

And so what is the question there?

The question is, take the component of v

in the direction of all the xi.

And then look for this v, which is doing this best for all xi,

which means I try to maximize the target function here,

such that I have to fulfill the side conditions

that the length of this thing is not larger than 1.

And if you want to optimize this thing,

then you can say an absolute value to the square

is the same as in the scalar product

is the same as the vector to the square.

And if I write this down and if I

do it correctly with the transpose matrix,

then I would see such a thing here.

I can rewrite it as we have done before.

And then you see it in such a matrix formulation here.

And to find such a vector, then is that nothing else?

Then find such a vector else here in this formulation.

And together with the side constraints,

you can say now I have a Lagrange problem, which

means optimization under an equation side condition.

So what is the optimization?

That is what we have written down here.

What is the sine condition?

Please have in mind this is a one-dimensional thing.

This is 1 minus v v transpose v transpose v.

This thing is equal to 1.

And so all together here, it's equal to 0.

And so you have a classical formulation of a Lagrange problem,

which means we know how to solve it.

The derivative of the Lagrange function down

to the variables we are interesting in

is going back to one of the preliminary slides here.

The first derivative of such a function description here

is 2 v transpose x x transpose.

And the first derivative of this thing here