jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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5DUQ3-Y_gX4	you need to multiply by a row vector.
5DUQ3-Y_gX4	Another way of thinking about it is you
5DUQ3-Y_gX4	need to take the dot product--
5DUQ3-Y_gX4	ALAN EDELMAN: The vector and the scalar out.
5DUQ3-Y_gX4	STEVEN G. JOHNSON: The vector and scalar out-- sorry.
5DUQ3-Y_gX4	You need to multiply it by a one row thing.
5DUQ3-Y_gX4	Another way of thinking about it is
5DUQ3-Y_gX4	that, if you have a linear operation that
5DUQ3-Y_gX4	takes a vector in and gives you a scalar
5DUQ3-Y_gX4	out, the only type of thing that does that is a dot product.
5DUQ3-Y_gX4	If you take a dot product of the vector, you get a scalar.
5DUQ3-Y_gX4	And that's the only linear operation
5DUQ3-Y_gX4	that gives you a scalar from a vector in some sense.
5DUQ3-Y_gX4	And so this is a dot product with some vector.
5DUQ3-Y_gX4	That vector must be pretty special.
5DUQ3-Y_gX4	And so we'll give it a name.
5DUQ3-Y_gX4	And we'll call that the gradient.
5DUQ3-Y_gX4	So I think this is going to be the gradient of f.
5DUQ3-Y_gX4	And this is the thing we take the dot product with
5DUQ3-Y_gX4	to get our scalar df.
5DUQ3-Y_gX4	So you can think of this in linear algebra terms.
5DUQ3-Y_gX4	So this dot product is the same thing
5DUQ3-Y_gX4	as multiplying by a transpose.
5DUQ3-Y_gX4	So this is the same thing as--
5DUQ3-Y_gX4	this is saying that f prime is really grad f transpose.
5DUQ3-Y_gX4	Or equivalently, f prime dx is the operation
5DUQ3-Y_gX4	of a dot product with a gradient.
5DUQ3-Y_gX4	This is going to be really powerful pretty soon
5DUQ3-Y_gX4	because it's going to also allow us to generalize gradients
5DUQ3-Y_gX4	to other kinds of vector spaces.
5DUQ3-Y_gX4	As long as we have a dot product and a scalar function,
5DUQ3-Y_gX4	you'll be able to define a gradient.
5DUQ3-Y_gX4	So if you have a scalar function of-- if this is something that
5DUQ3-Y_gX4	takes a matrix in and a scalar out, like a determinant,
5DUQ3-Y_gX4	pretty soon we're going to be able to take
5DUQ3-Y_gX4	the gradient of a determinant.
5DUQ3-Y_gX4	We'll be able to define what that means.
5DUQ3-Y_gX4	ALAN EDELMAN: Just to be clear, what
5DUQ3-Y_gX4	is on the other side of that equals sign
5DUQ3-Y_gX4	that you just wrote?
5DUQ3-Y_gX4	STEVEN G. JOHNSON: So, yes, the f--
5DUQ3-Y_gX4	yeah, I should write that.
5DUQ3-Y_gX4	That's the f prime.
5DUQ3-Y_gX4	Yeah.
5DUQ3-Y_gX4	So this is-- yeah.
5DUQ3-Y_gX4	I need to-- my equals--
5DUQ3-Y_gX4	let's see.
5DUQ3-Y_gX4	The gradient of f is the thing we take a dot product.
5DUQ3-Y_gX4	And here it's f prime of x is this.
5DUQ3-Y_gX4	ALAN EDELMAN: Exactly.
5DUQ3-Y_gX4	It's f prime of x.
5DUQ3-Y_gX4	And it's not df.
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Yeah.
5DUQ3-Y_gX4	It's not-- no dx.
5DUQ3-Y_gX4	Just the f prime by itself is a row vector.
5DUQ3-Y_gX4	That's the transpose of the gradient.
5DUQ3-Y_gX4	So now, that's the definition of the gradient.
5DUQ3-Y_gX4	It's only something we're usually
5DUQ3-Y_gX4	going to define for scalar functions of vectors.
5DUQ3-Y_gX4	And pretty soon, we'll be able to generalize that
5DUQ3-Y_gX4	to other kinds of vectors, but will still be a scalar.
5DUQ3-Y_gX4	And it's the thing you take the dot product
5DUQ3-Y_gX4	with of dx, of the differential, and the change in the input
5DUQ3-Y_gX4	with to get the change in the output, OK?
5DUQ3-Y_gX4	Yeah.
5DUQ3-Y_gX4	So we did-- Alan did the example of x transpose x.
5DUQ3-Y_gX4	Let's do another example just for fun.
5DUQ3-Y_gX4	So suppose f of x.
5DUQ3-Y_gX4	So x is going to be a vector.
5DUQ3-Y_gX4	Suppose that x transpose Ax where-- so this is x.
5DUQ3-Y_gX4	x here is going to have m components.
5DUQ3-Y_gX4	And so we're going to let A be an m by m matrix.
5DUQ3-Y_gX4	ALAN EDELMAN: And are you assuming asymmetric or--
5DUQ3-Y_gX4	STEVEN G. JOHNSON: No, I'm not.
5DUQ3-Y_gX4	I won't make it symmetric just for generality, OK?
5DUQ3-Y_gX4	And so this is going to be--
5DUQ3-Y_gX4	A is not going to be an input.
5DUQ3-Y_gX4	This is just going to be a constant matrix.
5DUQ3-Y_gX4	So when I do my d's, when I change x, A does not change.
5DUQ3-Y_gX4	OK.
5DUQ3-Y_gX4	So let me do it the long way first,
5DUQ3-Y_gX4	and then we'll try and derive some rules.
5DUQ3-Y_gX4	So let's do the long way--
5DUQ3-Y_gX4	long way, but still faster than doing it
5DUQ3-Y_gX4	component by component, faster than the 18.02 component
5DUQ3-Y_gX4	by component.
5DUQ3-Y_gX4	Because I could take the derivative of this with respect
5DUQ3-Y_gX4	to x1, with respect to x2, with respect to all the components,
5DUQ3-Y_gX4	and then build up the gradient.
5DUQ3-Y_gX4	It starts to become really awkward really quickly.
5DUQ3-Y_gX4	I know it can be tempting when you're faced with new problems
5DUQ3-Y_gX4	to fall back on what you know, right?
5DUQ3-Y_gX4	And that's not a bad strategy.
5DUQ3-Y_gX4	But we really want to encourage you to--
5DUQ3-Y_gX4	even though you know calculus really,
5DUQ3-Y_gX4	really well, you know how to take derivatives really, really
5DUQ3-Y_gX4	well, 18.02 and 18.01 style, we're
5DUQ3-Y_gX4	going to try and learn something new here,
5DUQ3-Y_gX4	a new way that can be really a lot more powerful besides
5DUQ3-Y_gX4	[INAUDIBLE].

5DUQ3-Y_gX4

you need to multiply by a row vector.

5DUQ3-Y_gX4

Another way of thinking about it is you

5DUQ3-Y_gX4

need to take the dot product--

5DUQ3-Y_gX4

ALAN EDELMAN: The vector and the scalar out.

5DUQ3-Y_gX4

STEVEN G. JOHNSON: The vector and scalar out-- sorry.

5DUQ3-Y_gX4

You need to multiply it by a one row thing.

5DUQ3-Y_gX4

Another way of thinking about it is

5DUQ3-Y_gX4

that, if you have a linear operation that

5DUQ3-Y_gX4

takes a vector in and gives you a scalar

5DUQ3-Y_gX4

out, the only type of thing that does that is a dot product.

5DUQ3-Y_gX4

If you take a dot product of the vector, you get a scalar.

5DUQ3-Y_gX4

And that's the only linear operation

5DUQ3-Y_gX4

that gives you a scalar from a vector in some sense.

5DUQ3-Y_gX4

And so this is a dot product with some vector.

5DUQ3-Y_gX4

That vector must be pretty special.

5DUQ3-Y_gX4

And so we'll give it a name.

5DUQ3-Y_gX4

And we'll call that the gradient.

5DUQ3-Y_gX4

So I think this is going to be the gradient of f.

5DUQ3-Y_gX4

And this is the thing we take the dot product with

5DUQ3-Y_gX4

to get our scalar df.

5DUQ3-Y_gX4

So you can think of this in linear algebra terms.

5DUQ3-Y_gX4

So this dot product is the same thing

5DUQ3-Y_gX4

as multiplying by a transpose.

5DUQ3-Y_gX4

So this is the same thing as--

5DUQ3-Y_gX4

this is saying that f prime is really grad f transpose.

5DUQ3-Y_gX4

Or equivalently, f prime dx is the operation

5DUQ3-Y_gX4

of a dot product with a gradient.

5DUQ3-Y_gX4

This is going to be really powerful pretty soon

5DUQ3-Y_gX4

because it's going to also allow us to generalize gradients

5DUQ3-Y_gX4

to other kinds of vector spaces.

5DUQ3-Y_gX4

As long as we have a dot product and a scalar function,

5DUQ3-Y_gX4

you'll be able to define a gradient.

5DUQ3-Y_gX4

So if you have a scalar function of-- if this is something that

5DUQ3-Y_gX4

takes a matrix in and a scalar out, like a determinant,

5DUQ3-Y_gX4

pretty soon we're going to be able to take

5DUQ3-Y_gX4

the gradient of a determinant.

5DUQ3-Y_gX4

We'll be able to define what that means.

5DUQ3-Y_gX4

ALAN EDELMAN: Just to be clear, what

5DUQ3-Y_gX4

is on the other side of that equals sign

5DUQ3-Y_gX4

that you just wrote?

5DUQ3-Y_gX4

STEVEN G. JOHNSON: So, yes, the f--

5DUQ3-Y_gX4

yeah, I should write that.

5DUQ3-Y_gX4

That's the f prime.

5DUQ3-Y_gX4

Yeah.

5DUQ3-Y_gX4

So this is-- yeah.

5DUQ3-Y_gX4

I need to-- my equals--

5DUQ3-Y_gX4

let's see.

5DUQ3-Y_gX4

The gradient of f is the thing we take a dot product.

5DUQ3-Y_gX4

And here it's f prime of x is this.

5DUQ3-Y_gX4

ALAN EDELMAN: Exactly.

5DUQ3-Y_gX4

It's f prime of x.

5DUQ3-Y_gX4

And it's not df.

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Yeah.

5DUQ3-Y_gX4

It's not-- no dx.

5DUQ3-Y_gX4

Just the f prime by itself is a row vector.

5DUQ3-Y_gX4

That's the transpose of the gradient.

5DUQ3-Y_gX4

So now, that's the definition of the gradient.

5DUQ3-Y_gX4

It's only something we're usually

5DUQ3-Y_gX4

going to define for scalar functions of vectors.

5DUQ3-Y_gX4

And pretty soon, we'll be able to generalize that

5DUQ3-Y_gX4

to other kinds of vectors, but will still be a scalar.

5DUQ3-Y_gX4

And it's the thing you take the dot product

5DUQ3-Y_gX4

with of dx, of the differential, and the change in the input

5DUQ3-Y_gX4

with to get the change in the output, OK?

5DUQ3-Y_gX4

Yeah.

5DUQ3-Y_gX4

So we did-- Alan did the example of x transpose x.

5DUQ3-Y_gX4

Let's do another example just for fun.

5DUQ3-Y_gX4

So suppose f of x.

5DUQ3-Y_gX4

So x is going to be a vector.

5DUQ3-Y_gX4

Suppose that x transpose Ax where-- so this is x.

5DUQ3-Y_gX4

x here is going to have m components.

5DUQ3-Y_gX4

And so we're going to let A be an m by m matrix.

5DUQ3-Y_gX4

ALAN EDELMAN: And are you assuming asymmetric or--

5DUQ3-Y_gX4

STEVEN G. JOHNSON: No, I'm not.

5DUQ3-Y_gX4

I won't make it symmetric just for generality, OK?

5DUQ3-Y_gX4

And so this is going to be--

5DUQ3-Y_gX4

A is not going to be an input.

5DUQ3-Y_gX4

This is just going to be a constant matrix.

5DUQ3-Y_gX4

So when I do my d's, when I change x, A does not change.

5DUQ3-Y_gX4

OK.

5DUQ3-Y_gX4

So let me do it the long way first,

5DUQ3-Y_gX4

and then we'll try and derive some rules.

5DUQ3-Y_gX4

So let's do the long way--

5DUQ3-Y_gX4

long way, but still faster than doing it

5DUQ3-Y_gX4

component by component, faster than the 18.02 component

5DUQ3-Y_gX4

by component.

5DUQ3-Y_gX4

Because I could take the derivative of this with respect

5DUQ3-Y_gX4

to x1, with respect to x2, with respect to all the components,

5DUQ3-Y_gX4

and then build up the gradient.

5DUQ3-Y_gX4

It starts to become really awkward really quickly.

5DUQ3-Y_gX4

I know it can be tempting when you're faced with new problems

5DUQ3-Y_gX4

to fall back on what you know, right?

5DUQ3-Y_gX4

And that's not a bad strategy.

5DUQ3-Y_gX4

But we really want to encourage you to--

5DUQ3-Y_gX4

even though you know calculus really,

5DUQ3-Y_gX4

really well, you know how to take derivatives really, really

5DUQ3-Y_gX4

well, 18.02 and 18.01 style, we're

5DUQ3-Y_gX4

going to try and learn something new here,

5DUQ3-Y_gX4

a new way that can be really a lot more powerful besides

5DUQ3-Y_gX4

[INAUDIBLE].