Let's say you're working on a cool image processing project and your goal is to build an algorithm that analyzes faces for emotions. It takes in a 256 pixel by 256 pixel grayscale image as its input and spits out an emotion as an answer. For example, if you passed in the following image, you'd expect the algorithm to label it as "happy."

Now this is all well and good, but before we're satisfied with this approach, let's take a step back and think about what this really means. A 256 by 256 grayscale image corresponds to an input vector of over 65,000 dimensions! In other words, we're trying to solve a problem in a 65,000-dimensional space. That's not a particularly easy thing to do, even for a computer! Not only are large inputs annoying to store, move around, and compute with, but they also give rise to some pretty serious tractability concerns.