Error bounds for high-resolution quantization with Rényi-α-entropy constraints

Abstract

We consider the problem of optimal quantization with norm ex- ponent r > 0 for Borel probability measures on R d under constrained Renyi-fi- entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the spe- cial cases fi = 0 (memory-constrained quantization) and fi = 1 (Shannon-entropy- constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter fi 2 (1 +r=d;1). For fi 2 (0;1 + r=d( we specify the asymptotical order of the op- timal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all fi 2 (0;1).