Visualization and Projection by Jesse Johnson.
From the post:
One of the common themes that I’ve emphasized so far on this blog is that we should try to analyze high dimensional data sets without being able to actually “see” them. However, it is often useful to visualize the data in some way, and having just introduced principle component analysis, this is probably a good time to start the discussion. There are a number of types of visualization that involve representing statistics about the data in different ways, but today we’ll look at ways of representing the actual data points in two or three dimensions.
In particular, what we want to do is to draw the individual data points of a high dimensional data set in a lower dimensional space so that the hard work can be done by the best pattern recognition machine there is: your brain. When you look at a two- or three-dimensional data set, you will naturally recognize patterns based on how close different points are to each other. Therefore, we want to represent the points so that the distances between them change as little as possible. In general, this is called projection, the term coming from the idea that we will do the same thing to the data as you do when you make shadow puppets: We project a high dimensional object (such as your three-dimensional hands) onto a lower dimensional object (such as the two-dimensional wall).
We’ve already used linear projection implicitly when thinking about higher dimensional spaces. For example, I suggested the you think about the three-dimensional space that we live in as being a piece of paper on the desk of someone living in a higher dimensional space. In the post about the curse of dimensionality, we looked at the three-dimensional cube from the side and saw that it was two-dimensional, then noted that a four-dimensional cube would look like a three-dimensional cube if we could look at it “from the side”.
When we look at an object from the side like this, we are essentially ignoring one of the dimensions. This the simplest form of projection, and in general we can choose to ignore more than one dimension at a time. For example, if you have data points in five dimensions and you want to plot them in two dimension, you could just pick the two dimensions that you thought were most important and plot the points based on those. It’s hard to picture how that works because you still have to think about the original five dimensional data. But this is similar to the picture if we were to take a two-dimensional data set and throw away one of the dimensions as in the left and middle pictures in the Figure below. You can see the shadow puppet analogy too: In the figure on the left, the light is to the right of the data, while in the middle, it’s above.
I hesitated over:
…then noted that a four-dimensional cube would look like a three-dimensional cube if we could look at it “from the side”
Barely remembering Martin Gardner’s column on visualizing higher dimensions and the illustrations of projecting a fourth dimensional box into three dimensions.
But the original post is describing a fourth dimensional cube in a fourth dimensional space being viewed “on its side” by a being that exists in fourth dimensional space.
How would you choose which dimensions to project for a human observer to judge?