Minimum Distortion Quantization with Specified Output Distribution

arXiv:2606.10458v1 Announce Type: cross Abstract: We derive the optimal quantizer of a real-valued random variable $W$ with distribution $P_W$ such that 1) the distribution of the quantization output $X$ that can take $k$ values follows any specified distribution $P_X$ over $\{1,\ldots,k\}$, and 2) the minimum mean squared error (MMSE) of estimating $W$ from $X$ is minimized. It is shown that the optimal quantizer takes the form $X=\sigma\big(F_{\sigma^{-1}(X)}^{-1}(F_W(W))\big)$, where $\sigma$