Manifold Learning - part 1

Background: How many dimensions?

When we talk of dimensions in casual conversation, we often recall high school geometry. A point has zero dimensions, a line segment has one dimension (length), a rectangle has two dimensions (length & width), and a block has three dimensions (length, width & height).

We can also think of dimensions as a tuple, or a set of numbers, and this set of numbers describe something. For example, you can think of color in terms of Red, Green, and Blue components. We can say color has three dimensions (R, G and B). The same color can be represented with a different set of numbers; Cyan, Yellow, Magenta, and blacK. This time, color has four dimensions (C,Y,M, and K). If we are consistent with our set of numbers, we can describe many things. eHarmony supposedly has 29 dimensions to describe each person. It simply means they use 29 numbers to describe a person, whatever those numbers are supposed to measure.

Now analysing three dimensions is straightforward. We can turn it into graphs and plots, and it is easy to visualize. Four dimensions, a little harder but doable (look up color solid or color space sometime). But, 29 dimensions? How about 19,200 dimensions? We need help for those.