Name: 5 6 次元削減の導入 12 01 (5 6 Dimensionality Reduction Introduction 12 01)
Uploaded: 2021-01-14T07:02:43.000Z
Duration: 12 min 2 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

基礎受動態

The topic we will discuss today is, is called dimensionality reduction.

様態の副詞

And the idea here is basically that we will learn about techniques that will

later become very handy when we will talk about recommender systems, and

in particular latent factor recommender systems.

So let me give you an idea of what the problem of dimensionality reduction

So basically our assumption is that we have a set of data points.

Think of them as points in a plane or points in a three-dimensional space.

And the idea is that these points are not just randomly scattered through the space,

but they, they li, lie in a subspace of it.

So for example, here, here I have two cases of this.

You could imagine that you have a set of data in a two-dimensional plane, but

the data is not only kind of randomly scattered through this plane,

but it it it is only scattered across a small subspace of it.

So for example in the first case, we have our we have the data points that are that

are embedded on this particular line so maybe a more better representation of

this data is not in this two-dimensional space but it's basically just

where where in the length of the line is, is a given data point.

Or, for example, in the second case, we have we, we are drawing a case where we

have points embedded in a three-dimensional space, but again, these

point, points are not randomly scattered through space, but basically, they are,

they are, they all lie on this single plane that is embedded in this space.

So basically the idea for axes can we go and discover such data in presentation.

So if I give you another clear set of data can we go identify what are the main

axes along original data is represented or embedded.

we have these 2 are an axis where all the data lies.

So our goal in some sense will be that we want to find a sub space

that effectively represents all the data in that we are given.

So, let me just give you a complete example, right.

So, our goal, in a sense, would be that we want to compress or

reduce the dimensionality or the size of the data representation.

So the way we can think of this is that we are given a big table with a, large number

of rows, let's say millions of rows, and also a large number of, of columns.

of this kind of table is that every row represents a different data point.

And every column represents a different coordinate or a dif, different dimension.

And our goal is that we take this set of data and

identify kind of more compact or fewer dimensional representations.

So in a sense, we would like to keep all the rows.

But we would like to shrink the number of columns.

While, while stoll, still preserve the richness of a da, of the data set.

So, for example, let's look at the the table that I have here.

I have, for example, a table where every row is a different customer and

every column is a different time of the day, where every entry stores how many.

But how many of particular transactions or

particular products need a particular customer to buy.

And for example what we see in this particular case is that even though we

have five different days so five different columns,

our data is not really in some sense five dimensional but it's only two dimensional.

What do I mean by this is that for example all the first four rows and

the first three columns, they're basically all multiplications of one another, right?

So since I have a set of customers that all buy products on

the first in the first three columns and they do nothing on the last.

Two and then I have another set of let's say, customers.

That they will make transactions over the weekends.

And they don't do anything over the week.

rather than representing every customer now with the with a set of five values.

I can, I can simply represent this data with a.

Two two coordinate vectors, plus a value of which,

in some sense, which dimension or which cluster it belongs to, right?

So for example, this matrix that I showed you is really two dimensional, where every

row is simply a multiplication of one of the, one of the two vectors of 1s and 0s.

us will be can we identify this kind of low low level of representation of data.

So let me explain a concept that will be very important for

So we are thinking that our data comes in the form of a matrix right.

So we can think of matrix basically as every line giving us,

giving us coordinates of a point in some d-dimensional space.

So we have our data point, we have some number of data points, and we have some

number of columns which is corresponds to the dimensionality of the data.

And now the question is, what is the real intrinsic dimensionality to that data set?

And the concept we need to explain is the concept of a rank of a matrix.

And we will say that the rank of a matrix A is simply the number of

You can see that the matrix A that has three rows and three columns.

Why's the rank of this matrix equal to 2?

Is because it has 2 linear, linearly independent rows in this case.

What do we notice for example is that, I can,

that the row number 3 is simply the sum of rows one and two.

of this matrix can be represented as a linear combination of rows one and two.

So in this case our matrix is really two dimensional.

Even I have a, I have data, in three dimensions.

I have three columns, this matrix is really two dimensional.

So how can we think about this is the following?

I can basically think that there are really like two basis vectors or

two coordinate vectors in my in my space first one corresponds to

the first row second one corresponds to the second row and then what I can do

now is I can represent every data point as a linear combination of these two vectors.

the first row can simply be represented as a vector of one and zero.

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5 6 次元削減の導入 12 01 (5 6 Dimensionality Reduction Introduction 12 01)

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