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In this lecture, we are going to continue our discussion of convolution by looking at a totally different

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perspective on how it works.

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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.

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If you want to move on and just learn about how to make CNN's.

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Something I mentioned very often is factorization.

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We like to victories operations and code because using Nampai functions is a lot more efficient than

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writing your own python for loops.

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The pattern you're looking for when you want to victories in operation is usually of the form of a product.

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A dot product is an element y's multiplication and then the summation of those results.

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So whenever you see the sum over I of items by, that's a dot product.

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This also applies to matrix multiplication, since if amber matrices, then the matrix multiplication

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of AB is just B.

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You'll notice that for convolution, we yet again have something similar.

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The only difference is that there are two summations.

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However, this is not really a relevant detail, since the outcome is the same instead of summing over

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one axis or something over two axes.

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But it's still an element y some an ad.

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The question is, why is the DOT products important?

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One definition of the DOT product other than that element, why some an ad is that it's the magnitude

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of a multiplied by the magnitude of B multiplied by the cosine of the angle between A and B.

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We sometimes call this the cosine similarity or cosine distance, depending on the sign that you use.

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So how does this work geometrically consider just the angle for a moment, if the angle between the

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two vectors is zero, then the cosine of that angle is one that's the maximum value of the cosine.

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Now, imagine that the angle between two vectors is 90 degrees, then the cosine of that angle is zero.

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Finally, imagine that the angle between the two vectors is 180 degrees.

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This is basically as far apart as possible, then the cosine of that angle is minus one, which is the

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minimum value of the cosine.

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So if you're just using raw cosine, then it's a similarity.

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The larger the number, the closer the two vectors are, the smaller the number, the further away they

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are, the maximum value.

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When amber parallel is one, the minimum value when and B are A. Parallel is minus one.

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And if A and B are orthogonal, then the cosine similarity is just zero.

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OK, so why is that important?

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Consider now how you would find it the cosine of the angle between two vectors.

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That's just 8b divided by the magnitude of a in the magnitude of B.

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So I just rearrange the equation that we had before.

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Now, let's compare this to another popular measurement, the Pearson correlation, the Pearson correlation

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is defined as what you see here.

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But notice how similar this is to cosine similarity.

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The only difference is that the Pearson correlation uses mean subtraction.

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And so now you have two hints that convolution is really correlation.

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The first one from before it was that we were actually doing what is called the cross correlation.

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And second, now you have that the DOT product is actually very similar to the Pearson correlation.

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So I hope that by this point, you are convinced that the dog products, while that seems like an abstract

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concept, can be thought of as a correlation measure, it tells me how correlated is the first thing

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with the second thing.

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If they are highly, positively correlated, then the dog products should be large and positive.

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That means two vectors pointing in nearly the same direction.

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If they are highly negatively correlated, then the dot product should be large and negative.

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That means the two vectors are pointing in nearly opposite direction.

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Finally, if the two vectors are orthogonal or at right angles, then the DOT product should be zero.

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The reason why this is important is you don't have to think of a filter as an abstract concepts.

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In fact, it's just a pattern finder.

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This actually makes the term filter make a lot more sense.

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It filters out everything not related to the pattern contained in the filter by sending them to zero

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and keeps everything that is related to the pattern.

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So what convolution is doing is it's passing this filter along each point on the original input image

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and sliding it along at each point.

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It asks, Is the pattern here is the pattern, here is the pattern here and so forth.

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Then it gives us a high number in the positions where the pattern is found.

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And then a small number where the pattern is not found.

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And thus, this is your first alternative perspective on convolution.

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It's just a sliding pattern finder that passes through an entire image looking for a particular pattern.