Minimum Distortion Quantization with Specified Output Distribution
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 $X=σ\big(F_{σ^{-1}(X)}^{-1}(F_W(W))\big)$, where $σ$ is the optimal permutation of $\{1,\ldots,k\}$ among all permutations to minimize the MMSE, and $F$ is the cumulative distribution function. When $P_W$ is uniform over an interval or $P_X$ is uniform over $\{1,\ldots,k\}$, the quantizer takes a simple form $X=F...
| Authors | Aolin Xu |
|---|---|
| Primary category | cs.IT |
| Published | 9 June 2026 |
| Last updated at source | 9 June 2026 |
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