High-resolution scalar quantization with Rényi entropy constraint

We consider optimal scalar quantization with $r$th power distortion and constrained Rényi entropy of order $α$. For sources with an absolutely continuous distribution the high rate asymptotics of the quantizer distortion has long been known for $α=0$ (fixed-rate quantization) and $\al pha=1$ (entropy-constrained quantization). For a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion for $α\in [-\infty,0)\cup (0,1)$. The achievability proof is based on finding (asymptotically) optimal quantizers via the companding approach, and is thus constructive.

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