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5DUQ3-Y_gX4
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This somehow is a source
of endless confusion.
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5DUQ3-Y_gX4
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Alan already talked
about it when
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he said, for
example, dx transpose
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x equals x transpose dx.
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That's a little easier to
understand because it's just
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a dot product.
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You can swap things,
a scalar product.
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So this is also an
instance of the same rule.
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But you know, it's a little
bit more complicated looking.
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It's the same idea.
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Because this is a number,
I can transpose it.
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And that swaps
everything around.
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And the transpose of a product--
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hopefully, you remember
from linear algebra
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that the transpose
of a product is
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the product of the
transpose in reverse order.
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So this term here,
this whole term,
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equals x transpose
A transpose dx.
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And what that allows
me to do is it allows
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me to combine the two terms.
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Now, this term and
this term, they
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both have a dx on the right.
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And so I can put that
over in the right.
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And I can put parentheses.
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5DUQ3-Y_gX4
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And I have two terms,
and both of them
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have an x transpose on the left.
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And the first term,
one of the terms,
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has an A. The other
term has an A transpose.
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5DUQ3-Y_gX4
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So that means this thing here
is our derivative, f prime.
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5DUQ3-Y_gX4
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Again, don't get that confused
with the differential.
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f prime times dx is df.
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That's the change in the output.
|
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That's the little
change in the output.
|
5DUQ3-Y_gX4
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f prime is the rate of change.
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5DUQ3-Y_gX4
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It's the thing
you operate on dx.
|
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This is a row vector.
|
5DUQ3-Y_gX4
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Notice that this
is a row vector.
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5DUQ3-Y_gX4
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ALAN EDELMAN: Could
I ask the class?
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5DUQ3-Y_gX4
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Then quickly tell me what is
the gradient of x transpose dx.
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5DUQ3-Y_gX4
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Anybody want to shout it out?
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5DUQ3-Y_gX4
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What is the gradient?
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AUDIENCE: You [INAUDIBLE]?
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ALAN EDELMAN: Right.
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5DUQ3-Y_gX4
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Do you want to say
it in its full glory?
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5DUQ3-Y_gX4
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AUDIENCE: OK.
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5DUQ3-Y_gX4
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A plus-- A plus A
transpose times x.
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5DUQ3-Y_gX4
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ALAN EDELMAN: Good, yup.
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5DUQ3-Y_gX4
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A plus A transpose times x
is the gradient, exactly.
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STEVEN G. JOHNSON: Right.
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5DUQ3-Y_gX4
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It's just the transpose
of this thing.
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5DUQ3-Y_gX4
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So we transpose this thing.
dx goes over on the right.
|
5DUQ3-Y_gX4
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This thing gets transposed.
|
5DUQ3-Y_gX4
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But this is symmetric,
so it equals itself.
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ALAN EDELMAN: So,
Steven, on the clock
|
5DUQ3-Y_gX4
|
here, we're already at 12:56.
|
5DUQ3-Y_gX4
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So you might want to kind of--
|
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STEVEN G. JOHNSON: OK.
|
5DUQ3-Y_gX4
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ALAN EDELMAN: --come
to a conclusion.
|
5DUQ3-Y_gX4
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STEVEN G. JOHNSON: Yeah.
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5DUQ3-Y_gX4
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So this is just revisiting
the notion of a gradient.
|
5DUQ3-Y_gX4
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And so next time, we're going
to continue so that next time,
|
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basically, we're going to
do 18.06 revisited part two.
|
5DUQ3-Y_gX4
|
And we're going to
have f is now going
|
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to be a vector function that
takes a vector of outputs
|
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and also takes a
vector of inputs.
|
5DUQ3-Y_gX4
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And so we have outputs in,
say, Rm inputs in, say, Rn.
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5DUQ3-Y_gX4
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I probably should have used n
just to be consistent before.
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5DUQ3-Y_gX4
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And then what we're going
to find is that then--
|
5DUQ3-Y_gX4
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you can almost do it right now.
|
5DUQ3-Y_gX4
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df has to be a
linear operator that
|
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takes a small
change in the input
|
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and gives you a small
change in the output.
|
5DUQ3-Y_gX4
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And so this has to have m
outputs, m components here.
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5DUQ3-Y_gX4
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It has to have n
components there.
|
5DUQ3-Y_gX4
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The only way you can get
a linear operator that
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takes n inputs and m
outputs is that this
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has to be an m by n matrix.
|
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ALAN EDELMAN: It has to
be expressible as an m
|
5DUQ3-Y_gX4
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by n matrix.
|
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STEVEN G. JOHNSON: Exactly.
|
5DUQ3-Y_gX4
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ALAN EDELMAN: But you don't
have to write down the matrix.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON:
Yes, that's right.
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5DUQ3-Y_gX4
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So this is our f prime
of x linear operator.
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5DUQ3-Y_gX4
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And this, if we write
it down as a matrix,
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5DUQ3-Y_gX4
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we call it the Jacobian.
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5DUQ3-Y_gX4
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ALAN EDELMAN: I
think what I'll do
|
5DUQ3-Y_gX4
|
next time is show my
favorite nonlinear
|
5DUQ3-Y_gX4
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operator on two-dimensional
space, which is
|
5DUQ3-Y_gX4
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hyperbolic operators on corgis.
|
5DUQ3-Y_gX4
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So you'll see hyperbolic
corgis on Friday if you come.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON: So probably
on Friday maybe we'll switch.
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5DUQ3-Y_gX4
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And I'll start with the first
half and then finish this up.
|
5DUQ3-Y_gX4
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And then Alan can
do the second half.
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5DUQ3-Y_gX4
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ALAN EDELMAN: OK, we
could do it that way.
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5DUQ3-Y_gX4
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OK.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON: Thanks, all.
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5DUQ3-Y_gX4
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Any questions at this point?
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5DUQ3-Y_gX4
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But let's--
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5DUQ3-Y_gX4
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ALAN EDELMAN: I'll
also stick around.
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5DUQ3-Y_gX4
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You can ask Steven
or, you know--
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