[MUSIK]

welcome back and today want to hello so remember this and this is very important

for the oral exam when I said the past 5 minutes okay and don't tell the others good so

let let's continue what we're currently doing is we

are not only looking into feature transforms but also

feature transforms that reduce the number of features or

the dimensionality of features according to a certain optimality

criterion and last week we have

stopped with a problem what is the sub

manifold we can project features on such that the spread of the

variance of the projected features is maximum I'll just remind you to

the figure I have drawn on the blackboard we had these point

features and we are looking at these two dimension point features

in a way that we are looking for a sub manifold

whoops where is my pen sorry that

we're looking for a sub manifold that

shows the maximum spread for instance this straight line if

we project all the features on this line the resulting 1-D

projection points they will have these points will have the maximum

variance the maximum spread on the sub manifold or if you

have points like these here let me show you

another example like these here

points like these a curved

sub manifold for instance could

be this one here where we project

the features on a curved surface and the points

on these curved projected points on the curved surface

they show a maximum spread they are smeared in a way

of to a certain well they are smeared such that the

distance of the projected point the mean distance is maximum okay

good and this here is a linear manifold that's an affine

linear function and this here is a curved manifold this one

is way harder we will look into that later first of

all we look at manifolds like this and what we

do is we search for feature transforms phi that map our

d dimensional feature to a feature x-prime where the dimension of

x-prime is way lower and the transformation phi should be selected in

a way that we have the maximum spread now I can search

in two ways and that's what I have pointed out last week already

I can search in the space of all possible transformations so I search in

the space where all the functions that map D dimensional vectors to

lower dimensional vectors let's say k dimensional vectors and I try to

find a function that maximizes the spread of the projected points

the second way of looking at that could be I look

at linear mappings that map our d dimensional feature to a

k dimensional feature by a linear mapping and the linear mapping

should be chosen in a way that the spread of the

transform features should be maximum now a linear transform is characterized

by a matrix so if I have to map d dimensional

features to k dimensional features I have a matrix with k