Reformulation of the objective

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Do curso por Imperial College London

Mathematics for Machine Learning: PCA

276 classificações

Imperial College London

Mathematics for Machine Learning: PCA

276 classificações

Curso 3 de 3 em Specialization Mathematics for Machine Learning

This intermediate-level course introduces the mathematical foundations to derive Principal Component Analysis (PCA), a fundamental dimensionality reduction technique. We'll cover some basic statistics of data sets, such as mean values and variances, we'll compute distances and angles between vectors using inner products and derive orthogonal projections of data onto lower-dimensional subspaces. Using all these tools, we'll then derive PCA as a method that minimizes the average squared reconstruction error between data points and their reconstruction. At the end of this course, you'll be familiar with important mathematical concepts and you can implement PCA all by yourself. If you’re struggling, you'll find a set of jupyter notebooks that will allow you to explore properties of the techniques and walk you through what you need to do to get on track. If you are already an expert, this course may refresh some of your knowledge. The lectures, examples and exercises require: 1. Some ability of abstract thinking 2. Good background in linear algebra (e.g., matrix and vector algebra, linear independence, basis) 3. Basic background in multivariate calculus (e.g., partial derivatives, basic optimization) 4. Basic knowledge in python programming and numpy Disclaimer: This course is substantially more abstract and requires more programming than the other two courses of the specialization. However, this type of abstract thinking, algebraic manipulation and programming is necessary if you want to understand machine learning algorithms.

Na lição

Principal Component Analysis

We can think of dimensionality reduction as a way of compressing data with some loss, similar to jpg or mp3. Principal Component Analysis (PCA) is one of the most fundamental dimensionality reduction techniques that are used in machine learning. In this module, we use the results from the first three modules of this course and derive PCA from a geometric point of view. Within this course, this module is the most challenging one, and we will go through an explicit derivation of PCA plus some coding exercises that will make us a proficient user of PCA.

Welcome to module 41:08

Problem setting and PCA objective7:45

Finding the coordinates of the projected data5:29

Reformulation of the objective10:25

Finding the basis vectors that span the principal subspace7:39

Conheça os instrutores

Marc P. Deisenroth
Lecturer in Statistical Machine Learning
Department of Computing

In the last video,

we determined the coordinates of the optimal projection with respect

to the orthonormal basis that spans our principal subspace.

Before we go on and determine the optimal basis vectors,

let's rephrase our loss function first.

I have copied over the results that we have so far.

So the description of our projected data point, our loss function,

the partial derivative of our loss function with respect to

our projected data point and the optimal coordinates that we found in the loss video.

Before we go on and determine the optimal basis Vectors,

let's rephrase our loss function.

This will make it much easier to find our basis vectors.

For this, let's have a closer look at the difference vector between

our original data point and our projected data point. So you can write.

So Xn tilde is given by equation A,

which is the sum over j = 1 to M beta Jn times bj.

If we now use the results for our Optimal beta jn parameters from here, we get,

this is j = 1 to M xn transpose times

bj times bj where we used D. Now,

we rewrite this in the following way.

This is just a scalar or a dot product in this particular case,

dot products are symmetric so we can swap the order,

and we can also move the scalar over here.

So what we end up with is bj times bj transpose times Xn

and this one we can write generally as j = 1

to M times bj times bj transpose Xn,

where we move the Xn out of the sum.

And if we look at this,

this is a projection matrix.

So, this means that Xn tilde is

the orthogonal projection of Xn onto the subspace spanned by the M basis vectors,

bj where j = 1 to M. Similarly,

we can write Xn as the sum j = 1 to

M of bj times bj transpose times Xn,

plus a term that runs from M +1 to D,

bj times bj transpose times Xn.

So, we write Xn as a projection onto

the principal subspace plus a projection onto the orthogonal complement.

And this term is the one that is missing over here.

That's the reason why Xn tilde is the approximation to Xn.

So if we now look at the difference vector between Xn tilde and Xn,

what remains is exactly this term.

So Xn minus Xn tilde is

the sum J = M + 1 to D of

bj times bj transpose times Xn.

So, now we can look at this displacement vectors

of the difference between Xn and its projection,

and we can see that

the displacement vector lies exclusively in the subspace that we ignore.

That means the orthogonal complement to the principal subspace.

Let's look at an example in two dimensions.

We have a data set and two dimensions represented by these dots and now we are

interested in projecting them onto the U1 subspace.

Well, we do this and then look at

the difference vector between the original data and the projected data,

we get these vertical lines.

That means they have no x component,

no variation in x.

That means they only have a component that lives in the subspace U2 which

is the orthogonal complement to U1 which is the subspace that we projected onto.

So, with this illustration,

let's quickly rewrite this in a slightly different way.

Going to write this as sum of J = M +1 to D of

bj transpose Xn times bj and we're going to call

this now equation E. We looked at

the displacement vector between Xn and it's

a orthogonal projection onto the principal subspace, Xn tilde.

And now we're going to use this to reformulate our loss function.

So, from equation B,

we get that our loss function is 1 over N times

the sum n = 1 to N of Xn minus Xn tilde squared.

So, this is the average squared reconstruction error and now we're

going to use equation E for the displacement vector here.

So we rewrite this now using equation E as 1 over N times the sum N

= 1 to capital N. And now we're going

to use inside that squared norm this expression here.

So we get the sum j = M + 1 to

D of bj transpose times Xn times bj squared.

And now we're going to use the fact that the bjs form an

orthonormal basis and this will greatly simplify this expression,

and we will get 1 over N times the sum n = 1 to capital N times

the sum J = M + 1 to D of bj transpose times Xn squared.

And now we're going to multiply this out

explicitly and we get 1 over N times the sum over

n times the sum over j times

bj transpose times Xn times Xn transpose times bj.

So, this part is now identical to this part.

And now I'm going to rearrange the sums.

So I'm going to move the sum over j outside.

So I'll have sum over

J = M + 1 to D times bj transpose.

So this is independent of n,

times 1 over N the sum

n = 1 to N of Xn times Xn transpose

and there's a bj from here missing times bj.

So I'm going to bracket it now in this way.

And what we can see now is that if we look very carefully,

we can identify this expression as the data covariance matrix S,

because we assumed we have centred data.

So the mean of the data is zero.

This means now we can rewrite our loss function using the data covariance matrix,

and we get that our loss is the sum over j = M + 1 to D of bj

transpose times S times bj and we can

also use a slightly different interpretation

by rearranging a few terms and using the trace operator.

So, we can now also write this as the trace of the sum of j = M + 1 to D

of bj times bj

transpose times S and we

can now also interpret this matrix as a projection matrix.

This projection matrix takes

our data covariance matrix and project it onto

the orthogonal compliment of the principal subspace.

That means, we can reformulate the loss function as

the variance of the data projected onto the subspace that we ignore.

Therefore, minimising this loss is equivalent to minimising

the variance of the data that lies in

the subspace that is a orthogonal to the principal subspace.

In other words, we are interested in retaining

as much variance after projection as possible.

The reformulation of the average squared reconstruction error in terms of

the data covariance gives us an easy way

to find the basis vector of the principal subspace,

which we'll do in the next video.

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