Asymptotic Equipartition Property - Category Theory

Category Theory

A category theoretic definition for the equipartition property is given by Gromov Given a sequence of Cartesian powers of a measure space P, this sequence admits an asymptotically equivalent sequence of homogenous measure spaces (i.e. all sets have the same measure; all morphisms are invariant under the group of automorphisms, and thus factor as a morphism to the terminal object) .

The above requires a definition of asymptotic equivalence. This is given in terms of a distance function, giving how much an injective correspondence differs from an isomorphism. An injective correspondence is a partially defined map that is a bijection; that is, it is a bijection between a subset and . Then define

where denotes the measure of a set S. In what follows, the measure of P and Q are taken to be 1, so that the measure spaces are probability spaces. This distance is commonly known as the earth mover's distance or Wasserstein metric.

Similarly, define

|\log P:Q|_\pi = \frac {\sup_{p\in P^'}|\log p - \log \pi(p)|} {\log \min \left(|set(P)|,|set(Q)|\right)}

with taken to be the counting measure on P. Thus, this definition requires that P be a finite measure space. Finally, let

A sequence of injective correspondences are then asymptotically equivalent when

\mbox{dist}_{\pi_N}(P_N,Q_N) \to 0 \quad\mbox{ as }\quad N\to\infty

Given a sequence that is asymptotically equivalent to, the entropy H(P) of P may be taken as

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