What we mean by regularity

I’ve spoken before of regularity, but haven’t defined it exactly. But before going into that, let’s first consider the intuitive notion that comes to mind. By regularity we mean something exhibiting pattern, repetition, invariance. We say something is regular if it follows a rule. In fact, the word’s etymology matches this, as regular is derived from the latin regula, rule. Repetition and invariance result from the continued applicability of the rule, over time and/or space, to that which is regular. For example

1, 3, 5, 7, 9, 11, 13, 15….

we say this sequence is regular because it follows a rule. The rule is

each number is the result of adding 2 to the number before it

As per our scheme above, the rule is applicable throughout the sequence, the +2 difference repeats, and it is invariant.

This way of looking at regularity matches the language we’ve used previously when defining the key aspect of learning as the extraction of generally applicable knowledge from specific examples. In this case the specific examples would be any subset of the sequence, the general case is the sequence in its entirety, and the extracted knowledge is the rule “+2”.

We can take this further and try to formalize it by realizing one consequence of the repetition characteristic. And it is that something that repeats can be shortened. The reason is simple, if we know the rule, we can describe the entire object by just describing the rule. The rule, by continued repetition, will reproduce the object up to any length. We can use the example before, and note how the sequence can be described succinctly as

f(x) = 2x + 1

which is much shorter than the sequence (which can be infinite in fact). So we can think of the rule as a compression of the object, or from the other point of view, the object is the expansion of the rule. Here’s another example

In this case, the object is a fractal, which can be described graphically by a potentially infinite set of points. The level of detail is infinite in the sense that one can zoom-in to arbitrary levels without loss of detail. This is why the description at a literal level (ie pixels) is infinitely long. However, like all fractals, the Mandelbrot set can be compactly described mathematically. So we say the set is highly regular by virtue of the existence of a short description that can reproduce all its detail. Here’s the short (formal) description for the Mandelbrot set In mathematics the length of what we have called short description is known as Kolmogorov complexity, or alternatively, Algorithmic Information Content. It is a measure of the quantity of information in an object, and is the inverse of regularity as we have discussed it here. We say that something with a comparably low AIC exhibits regularity as it can be compressed down to something much shorter.

I’ll regularly return to the concept of regularity as it is a fundamental way to look at pretty much everything, and is thus a very deep subject.

 For lack of a better word, I’m using the word object to refer to that which can house regularity, which is basically anything you can think of

 Note that this is not the only way to look at regularity, but rather one of two main formalizations. What we have seen here is the algorithmic approach to complexity (and regularity), but there is also a statistical view that is more suited to objects that do not have a fixed description, but rather a statistical one.