A Connection between Good Rate-distortion Codes and Backward DMCs
(1307.7770)Abstract
Let $Xn\in\mathcal{X}n$ be a sequence drawn from a discrete memoryless source, and let $Yn\in\mathcal{Y}n$ be the corresponding reconstruction sequence that is output by a good rate-distortion code. This paper establishes a property of the joint distribution of $(Xn,Yn)$. It is shown that for $D>0$, the input-output statistics of a $R(D)$-achieving rate-distortion code converge (in normalized relative entropy) to the output-input statistics of a discrete memoryless channel (dmc). The dmc is "backward" in that it is a channel from the reconstruction space $\mathcal{Y}n$ to source space $\mathcal{X}n$. It is also shown that the property does not necessarily hold when normalized relative entropy is replaced by variational distance.
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