Complexity Explorer Santa Few Institute

Introduction to Renormalization

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7.1 Conclusion: Keeping the things that matter » Quiz Solution

1. What was unusual about the Cellular Automaton renormalization example?
A. we were able to find an exact coarse-graining, where the diagram commuted.
B. the CAs were non-renormalizable.
C. we solved simultaneously for the coarse-graining operation and the model that described the emergent dynamics at that coarse-graining.
D. in the middle of the video, Stephen Wolfram walked across the screen wearing a Gorilla suit.

Answer: (C). In all the other cases, we specified ahead of time the coarse-graining we wanted to do, and then tried to figure out what happened to the model. In the CA case, we put some restrictions on the coarse-graining (eliminating the trivial ones), but otherwise left things free. (B) is interesting: in some cases, Israeli and Goldenfeld could not find a g and P pair that worked on a particular scale. This does imply that some of the rules are non-renormalizable. But many are renormalizable, and for every model, at sufficiently large coarse-graining scales, it turned out they could. Meanwhile, (A) we've seen a couple of times -- in the Markov Chain renormalization, as well as the Krohn-Rhodes theorem example (but not, for example, in the Ising model case).


2. What does rate-distortion theory do for you?

A. it tells you how to trade off the cost of missing a fine-grained feature when you coarse-grain your data (on the one hand) with the benefits you get from the lower cost of information gathering required by the coarse-graining function, given a conversion factor between the two costs.
B. it specifies a unique coarse-graining given a particular utility function.
C. it eliminates the need for a coarse-graining operation, by replacing it with a utility function.
D. it measures the extent to which a coarse-graining produces an overly complex model.

Answer (A). The closest answer to the correct one is (B). However, while it's true that rate-distortion theory requires that you specify a utility function ("distortion function"), that's not enough to specify a coarse-graining -- you also have to specify the tradeoff parameter beta, that tells you how expensive it is to gather information relative to how expensive it is to make mistakes.