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In the previous video, we talked

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about how to use back propagation

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to compute the derivatives of your cost function.

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In this video, I want

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to quickly tell you about one implementational detail of

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unrolling your parameters from

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matrices into vectors, which we

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need in order to use the advanced optimization routines.

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Concretely, let's say

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you've implemented a cost function

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that takes this input, you know, parameters

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theta and returns the cost function and returns derivatives.

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Then you can pass this to

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an advanced authorization algorithm by fminunc

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and fminunc

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isn't the only one by the way.

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There are also other advanced authorization algorithms.

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But what all of them

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do is take those input

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pointedly the cost function,

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and some initial value of theta.

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And both, and these

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routines assume that theta and

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the initial value of theta, that

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these are parameter vectors, maybe

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Rn or Rn plus 1.

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But these are vectors and it

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also assumes that, you know, your cost

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function will return as

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a second return value this

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gradient which is also Rn

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and Rn plus 1. So also a vector.

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This worked fine when we

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were using logistic progression but

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now that we're using a neural

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network our parameters are

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no longer vectors, but instead

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they are these matrices where for

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a full neural network we would

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have parameter matrices theta 1, theta 2, theta 3

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that we might represent in Octave

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as these matrices theta 1, theta 2, theta 3.

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And similarly these gradient

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terms that were expected to return.

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Well, in the previous video we

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showed how to compute these

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gradient matrices, which was

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capital D1, capital D2,

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capital D3, which we

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might represent an octave as matrices D1, D2, D3.

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In this video I want

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to quickly tell you about the

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idea of how to take

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these matrices and unroll them into vectors.

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So that they end up

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being in a format suitable for

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passing into as theta here off for getting

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out for a gradient there.

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Concretely, let's say we

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have a neural network with one

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input layer with ten units,

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hidden layer with ten units

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and one output layer with

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just one unit, so s1

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is the number of units in layer one

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and s2 is the

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number of units in layer two, and s3 is a number

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of units in layer three.

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In this case, the dimension of

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your matrices theta and

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D are going to be

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given by these expressions.

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For example, theta one

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is going to a 10 by 11 matrix and so on.

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So in if you want

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to convert between these matrices.

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vectors.

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What you can do is take

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your theta 1, theta

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2, theta 3, and write this

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piece of code and this will

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take all the elements of

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your three theta matrices and

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take all the elements

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of theta one, all the

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elements of theta 2, all the

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elements of theta 3,

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and unroll them and put

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all the elements into a big long vector.

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Which is thetaVec and similarly

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the second command would take

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all of your D matrices and

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unroll them into a big

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long vector and call them

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DVec.
And finally

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if you want to go back from

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the vector representations to the matrix representations.

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What you do to get back

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to theta one say is take

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thetaVec and pull

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out the first 110 elements.

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So theta 1 has 110

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elements because it's a

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10 by 11 matrix so that

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pulls out the first 110 elements

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and then you can

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use the reshape command to reshape those back into theta 1.

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And similarly, to get

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back theta 2 you pull

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out the next 110 elements and reshape it.

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And for theta 3, you pull out

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the final eleven elements and run

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reshape to get back the theta 3.

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Here's a quick Octave demo of that process.

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So for this example

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let's set theta 1 equal

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to be ones of 10 by

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11, so it's a matrix of all ones. And

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just to make this easier seen,

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let's set that to be 2

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times ones, 10 by

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11 and let's

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set theta 3 equals 3

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times 1's of 1 by 11.

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So this is 3

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separate matrices: theta 1, theta 2, theta 3.

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We want to put all of these as a vector.

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ThetaVec equals theta

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1; theta 2

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theta 3.

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Right, that's a colon

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in the middle and like so

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and now thetavec is

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going to be a very long vector.

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That's 231 elements.

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If I display it, I find

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that this very long vector with

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all the elements of the first

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matrix, all the elements of

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the second matrix, then all the elements of the third matrix.

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And if I want to get back

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my original matrices, I can

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do reshape thetaVec.

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Let's pull out the first 110

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elements and reshape them to a 10 by 11 matrix.

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This gives me back theta 1.

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And if I then pull

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out the next 110 elements.

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So that's indices 111 to 220.

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I get back all of my 2's.

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And if I go

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from 221 up to

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the last element, which is

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element 231, and reshape to

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1 by 11, I get back theta 3.

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To make this process really concrete,

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here's how we use the unrolling

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idea to implement our learning algorithm.

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Let's say that you have some

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initial value of the parameters

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theta 1, theta 2, theta 3.

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What we're going to do

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is take these and unroll

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them into a long vector

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we're gonna call initial theta to

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pass in to fminunc

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as this initial setting of the parameters theta.

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The other thing we need to do is implement the cost function.

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Here's my implementation of the cost function.

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The cost function is going to

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give us input, thetaVec,

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which is going to be all

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of my parameters vectors that in

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the form that's been unrolled into a vector.

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So the first thing I'm going to

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do is I'm going to use

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thetaVec and I'm going to use the reshape functions.

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So I'll pull out elements from

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thetaVec and use reshape

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to get back my

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original parameter matrices, theta 1, theta 2, theta 3.

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So these are going to be matrices that I'm going to get.

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So that gives me a

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more convenient form in which

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to use these matrices so that I

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can run forward propagation and

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back propagation to compute my

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derivatives, and to compute my cost function j of theta.

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And finally, I can then

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take my derivatives and unroll

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them, to keeping the elements

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in the same ordering as I did when I unroll my thetas.

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But I'm gonna unroll D1, D2,

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D3, to get gradientVec

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which is now what my cost function can return.

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It can return a vector of these derivatives.

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So, hopefully, you now have

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a good sense of how to

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convert back and forth between

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the matrix representation of the

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parameters versus the vector representation of the parameters.

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The advantage of the matrix

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representation is that when

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your parameters are stored as

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matrices it's more convenient when

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you're doing forward propagation and

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back propagation and it's easier

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when your parameters are stored as

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matrices to take advantage

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of the, sort of, vectorized implementations.

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Whereas in contrast the advantage of

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the vector representation, when you

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have like thetaVec or DVec is that

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when you are using the advanced optimization algorithms.

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Those algorithms tend to

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assume that you have

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all of your parameters unrolled into a big long vector.

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And so with what we just

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went through, hopefully you can now quickly

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convert between the two as needed.