An image is a point in space

by the way, just for you, so what we're doing right now, do you see what
we're doing here? Yeah, so we're taking the two by two image, we're
unwrapping it, that's called vectorizing.
We're turning it into a row vector and then we're stacking different images
on top of each other as row vectors.
So that's usually- Oh, you're stacking that on top of each other.
Yeah, yeah. Okay, so- So this row represents the first
image and this row represents the second image, right?
And we see we have one, zero, zero,
one,
zero, one, one, like that.
But the point is that this
object is a single point in
four dimensions, just like each of these objects are single
points in two dimensions.
I see. So you can take that and roll with it and you can just
make big images, right?
250, you know, 64 by 64 images.
You can unwrap them into 4,096 coordinates.
into a single point within space.
And you can say that 4096 length vector is a single point in
4096 dimensions. So here's a question for you.
So the current photos that are being sent back from the James Webb- Yeah.
Is that- […] huge.
So that's a single point.
Even that alone- Yeah, yeah, yeah, yeah, anything.
It's a single point in ridiculously high dimensional space.
Yeah, that'd be crazy.
Yeah. It's so much data.
Yeah. Those […] are massive.
Yeah, yeah, yeah. So I mean, it would be an absurdly high space.
Right. But the point is, it's still a- It's still a point- It's still just a
point within-
It's still just a point within- Oh, I see, I see, I see.
Right? So- Because you can always just unwrap it the same way you did with
the small- So is the dimensionality defined by the amount that you unwrap?
Exactly. So it's like 0, 1 to n.
Mm-hm. And so like this n being the
dimensionality. Exactly, exactly.
Oh, that's interesting.
So that's 100 points in however many- Right.
If they're 64 by 64, that's 100 points and those points live in
how many dimensions?
If each image is 64 by 64.
That's 64 dimensions.
No. Well, wait, did you say- It's a- Each one is 64 by 64.
Oh, then it's going to be 64 to the 64th power.
No. No? What would it be?
64 to the 64th power is 64 times 64 times 64 times
64 times 64 times 64 times 64 times 64 times 64 times 64 times 64 times 64
times 64 times 64 times 64 times 64 times 64 times- So it's going to be 64
times 64. Exactly. I see.
So let's be really clear about the rows are observations and the
columns are information about those
observations. So that's always how data tables work.
So? But at least that's always how we're going to think about data tables.
So observations right here.
Yep. And then this is information.
Yep. Also called the- In machine learning we would call those the features.
Yeah. You could also call it the dimension.
But dimension, I don't really like dimension because, right, when we talked
about that tensor over there, we're saying that tensor is three-dimensional
and that's kind of a different thing.
So let's just stick with features.
Because there's a few different ways to interpret dimension and I don't want
to get into that. Because that can get a little confusing.
For whoever might be watching this, I apologize for my handwriting.
So okay great so observations are the rows and the features
are the color. This is a good exercise.
So now say these are RGB.
So now how does that change things?
What do we do if they're RGB?
Oh interesting, okay.
So if they're RGB we then need...
Is that... Where's the […]...
I'll just grab this one.
So if it's RGB then we actually need it to dictate
not to zero to one
it would actually need to...
Okay.
It would be a matrix of
the red green blue champ.
I think what it would be is it would be the first image which would be R.
You'd put all the R information.
So say we'll just put some arbitrary numbers in here just to
do some […]. Yeah. Um,
one, then zero,
zero, one. Mm-hmm. So the way it would roll out is it
would still be a single row but then it would be zero,
one, zero,
one, zero,
one, zero, one. And that would be your image.
Wait, so where'd you get the zero, one, zero, one, zero, one, zero, one from?
I'm just dictating an arbitrary value in there.
Okay. Just so I could explain how it would be packaged
into a matrix. Got it.
So, um,
yeah. So, I think this brings an important
consideration into account that there's a lot of different ways to represent
information. Yeah. Um, people actually do do this in practice a lot
of the time, right? Is this?
So that's, they'll concatenate the red, green, and blue channels.
Is this, like, it's obviously viable but is it,
I feel like you, it's not that you're gonna lose
information. It gets muddy to read at a point.
Yeah, yeah, but that's fine, right?
I mean, um, how you organize it, it,
if you, if you organize something differently, then it's just
a different organizational structure a lot of the time, right?
Like, you can organize it however you want as long as you don't lose
information and most of the time it'll be the same result.
So, you'll just, you'll just have three matrices, right?
Right. You'll have one green matrix, one red matrix, and one
blue matrix. Um, and then, uh, the nice thing is that in
practice, you know, with something like, uh, PyTorch, which, um, or
NumPy, which is what, you know,
ML practitioners would actually use in practice to store data.
Um, you have this three-dimensional array, so, right?
Right, so you could store data like, um,
uh, you could store data into one array, but it would be the first dimension
is the stacks, uh, of, of, of,
think of a cube. Okay.
Right? And stacked on top of each other are, um,
uh, are images. Okay.
Um, and then, um,
sorry, it's a cube, um.
Do you want to draw it real quick?
Yeah, yeah, yeah, yeah.
It doesn't actually matter how you organize it, so I would say it
would be like, uh, say you have two, uh,
so this would be your
R layer, this would be your
B layer, and this would be your G layer.
Mm-hmm. Like, like this, right?
Yeah, I see. Like this, right?
And then, in the R layer, you would have, um, you know, your
first 0, 1, 0. Um, this would be your first
image, and then you would have a second image here,
and so on, right? So you would vectorize it the way that we talked about, but
then you would just stack the red, the blue, and the green.
So that is the same data structure as that 2D thing, so,
so this is literally,
imagine this, but then there's, like, layers of that.
Yeah. Right? So, like, you take this plane, and then you just add two more
planes.