Okay, so let me give you just a few more examples, Michael, of how PCA and ICA differ. I'm going to do this mainly by talking about certain things that I see each method does. This is really just for your edification, but I think it really helps to think about those underlying models and how they differ by considering how they react differently to different kinds of data. So here are a couple of examples: The first example we covered right away was the blind source separation problem. ICA was designed to solve this problem and, in fact, it does an excellent job at solving it. On the other hand, PCA does a terrible job at solving the blind source separation problem. This is because PCA assumes a Gaussian distribution of blind source in it, which is actually directional. To explain further, I drew a matrix before where we had features in one direction and samples of those features (sound time samples) in another direction. For PCA, it doesn't matter whether I give you this matrix or the transpose of this matrix, as it ends up finding the same answer. This makes sense because it's just finding a new rotation of the data and these rotations are effectively just rotations of each other if you think about it geometrically in space. However, ICA gives you completely different answers depending on whether you give it this matrix or the transpose of this matrix. ICA is highly directional, whereas PCA is much less directional. ICA, because of its fundamental assumption, ends up generating some really cool results. Let me give you another example, similar to the blind source separation problem. In fact, I'll give you two examples. Imagine you had a bunch of inputs of faces. If you applied PCA to these faces,...
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