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In this lecture, we are going to continue our discussion on alternative perspectives of convolution.

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Again, this is very helpful for understanding convolution, but it doesn't teach you anything new mechanically.

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So this lecture is optional if you want to just move on and learn about how to make science.

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So in this lecture, what were you going to talk about as the equivalence of convolution and a matrix

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multiplication?

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In other words, in what scenario are they actually the same thing?

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Let's do an example.

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This is easier to see if we only do a one dimensional conversation.

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So let's first go over how that would work.

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This should be easy since two dimensional convolution is just a generalization of this.

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So let's start with a one dimensional image.

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A equals A1, A2 a three eight four.

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We also have filter w equals W1 W2.

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Then the convolution output will be B, which is a convulsed with W, and the elements are A1 times

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one plus a two times W two, a two times W one plus a three times W two and a three times W one plus

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four times W two.

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So in general, we can write one dimensional convolution as the sum from I prime equals one up to K

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of eight I plus I prime times W of I prime.

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You'll notice that this is the same equation we had for two dimensional convolution, just with the

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second index dropped.

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Using this, it's pretty easy to see how we would implement the same thing using matrix multiplication.

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What we've done here is we created a matrix where we repeat the filter along each row, but on each

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row we shift it to the right by one space.

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Now I recommend you do this by hand so you can see that it works.

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If you multiply the matrix by the original input vector A., You get the correct output of the convolution.

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So what's the lesson here is that by repeating the same filter again and again inside a matrix, we

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can implement convolution without actually doing convolution.

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Instead, we can just use matrix multiplication.

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But there is another problem with this particular method of doing matrix multiplication, and this is

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that the equivalent matrix repeats the filter multiple times.

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So the result is that the Matrix takes up a lot more space than the original filter, which was just

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a two element array originally.

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In other words, while this was a helpful perspective, you don't necessarily want to implement convolution

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this way.

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Instead, it's helpful to look at this from the opposite direction.

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What if there are instances where instead of doing a faux matrix multiplication, we can replace it

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with convolution?

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This is the idea behind parameter sharing or weight sharing.

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Think of it this way inside a neuron that we are constantly doing.

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This operation equals W Transpose X.

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Here is the output activation and X is the input feature.

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Vector W is the weight matrix.

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Well, what if instead of a full weight matrix, we just had the same two weights repeating over and

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over again, then we have less parameters are known that work takes up less memory and ram, and thus

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we can make the computation more efficient.

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In other words, convolution saves both space and time by using less weights.

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Why might we want to do this?

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Consider what we did in the previous section where we had a fully connected neural network looking at

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a single image.

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Luckily, those images were just grayscale images of size 28 by 28, which gives us 784 features.

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But what if we had a slightly larger image and had color?

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The CFR 10 dataset is one such example.

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Those images are 32 by 32 by three.

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In this case, we have three thousand seventy two features.

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That's quite a large increase from 784.

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And keep in mind, a 32 by 32 image is quite modest in size, to say the least.

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Modern CNN, such as Viji look at images of size 224 by 224.

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If you use a full weight matrix, then you would have one hundred fifty thousand five hundred twenty

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eight features.

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Suppose you're looking at an image with modern HD resolution that's twelve eighty by 720.

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In this case, you would have about 2.8 million features.

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As you can imagine, this is much too large for a neural network.

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But there's another good reason to use change, which is that you don't need different weights for each

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part of the neural network.

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Remember that convolution is actually correlation and the filter is really a pattern finer.

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In this scenario, we actually want the same filter to look at all locations on the image.

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This is the idea behind translational and variance.

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Suppose we are building a dog versus cat, recognize it.

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Here is an image containing a cat.

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Here's another image containing a cat.

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But what's the difference between these two images?

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The only difference is that the cat is in a different position.

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Now, if we use the fully connected or dense neural network, we would have to learn the weights for

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each of these positions separately.

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And on top of that, this nerve network won't generalize well because if we come across the same cat

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but in a new position, the neural network would have failed to recognize it.

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So in fact, it's better to have a shared pattern finder because this pattern finder looks at all locations

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on the image.

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You don't need to learn the weights to look at a cat at every single possible point, which would actually

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be infeasible.