# Convolution

In common usage, ‘convoluted’ means “complex”,”hard to follow”,”unclear”. So the main reson to use ‘convoluted’ is for its emotional nuances. But I’ll argue (in the near future) that the full emotional power of language is overkill and makes expressing precise logical meaning quite difficult in practice. So as usual we turn to math for a better meaning.

In math, convolution is a peculiar operation that can be thought of as smearing two functions over their input. Conversely, deconvolution reverses the process, finding two functions from a smeared one. It’s often used in image/signal processing, where convolution is some kind of blur and deconvolution tries to sharpen. It’s used a little more generally for things like network deconvolution, where you attempt to direct effects when you’re only able to measure pairwise effects, which may be a direct effect, a transitive chain of 2, of 3, etc.

So let’s use our wonderful abstracting brains to distil the commonality:

DEFINITION: Convolution is when the distinction between multiple independent factors becomes obscured through some interaction. Deconvolution is the intellectual/computational process of dissecting a convoluted result into its factors, and determining how they convolute. Pseudomathematically, a convoluted thing is a thing like $C \overset{f}{\cong} C_1 + C_2 + C_3 + ...$. Deconvolution is finding both the right hand side (the factors) AND the function f, the “how”.

(Note: We could use “conflation” here and it would be closer to the original english meaning. I choose “convolution” instead so that the technical distinction is clear. The image here should be of an object “sitting below” its confounding factors. If this ‘misuse’ of language bothers you, feel free to interpret it as a frequent typo of mine – it doesn’t matter much anyway, as the definition will always be linked when using a “technical” term)

EXAMPLE: When statistically evaluating cause and effect, we often use correlation as a surrogate. But if two events A,B, have a 0.5 correlation, We could have

$A \overset{0.5}{\to} B$, or $A \overset{0.5}{\leftarrow} B$, or $A \overset{\sqrt{0.5}}{\leftarrow} C \overset{\sqrt{0.5}}{\to} B$ or $A \to \ldots \to B$ or.. you get the idea

Convolution is a big problem that seems to go unnoticed in human thinking (perhaps because there was no word for it in common usage :D), so I’ll be using it as a platform for many more posts in the near future. I started writing them and realized they all had the same braindump preamble, so I factored it out!