Maps and Adaptors

by Carson Reynolds

A map provides a formalization that describes how adaptation might operate. Here an object might be a space instead of a point, as would be the case with magnification.

It seems possible to describe a genetic algorithm as the generation of a set of maps which takes two members of the set a, the output of a random generator, and combines them to recover a new member of set a. Negative feedback would be describable in a 1D space as the production of a map conditioned on current input voltage. Simulated annealing takes the cooling schedule, the current state of the system, and random input to produce a map to the t+1 state. Newton’s method takes the first derivative at a point to recover the map to the next input of iteration.

It might be interesting to categorize different classes of adaptive algorithms by the properties of their map. An adaptive algorithm that entails a conformal mapping would have startling different behavior form another that constantly discards information.

A good next step would involve the description of different adaptive algorithms in terms of the attributes of their map production. It also points toward a clear relationship between the development of dynamics and the development of adaptive algorithms. In particular, I think Poincare’s use of iteration as documented in Alexander’s History of complex dynamics would help establish this linkage.