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Phase Transitions in Rate Distortion Theory and Deep Learning · Around

2020

Phase Transitions in Rate Distortion Theory and Deep Learning

Grohs, Philipp, Klotz, Andreas, Voigtlaender, Felix

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Rate distortion theory is concerned with optimally encoding a given signal class $\mathcal{S}$ using a budget of $R$ bits, as $R\to\infty$.

We say that $\mathcal{S}$ can be compressed at rate $s$ if we can achieve an error of $\mathcal{O}(R^{-s})$ for encoding $\mathcal{S}$; the supremal compression rate is denoted $s^\ast(\mathcal{S})$.
Given a fixed coding scheme, there usually are elements of $\mathcal{S}$ that are compressed at a higher rate than $s^\ast(\mathcal{S})$ by the given coding scheme; we study the size of this set of signals.
We show that for certain "nice" signal classes $\mathcal{S}$, a phase transition occurs: We construct a probability measure $\mathbb{P}$ on $\mathcal{S}$ such that for every coding scheme $\mathcal{C}$ and any $s >s^\ast(\mathcal{S})$, the set of signals encoded with error $\mathcal{O}(R^{-s})$ by $\mathcal{C}$ forms a $\mathbb{P}$-null-set.

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