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 $\sigma$ is the optimal permutation of $\{1,\ldots,k\}$ among all permutations to minimize the MMSE, and $F$
This is a fundamental research paper in information theory and signal processing, representing ongoing academic work in the field.
While foundational, this specific academic result on optimal quantization with specified output distribution is not immediately relevant to strategic readers.
This research contributes to the theoretical understanding of data compression and estimation, but does not represent a near-term change in application or technology.
Improved theoretical understanding of signal quantization limits.
Potential for marginal long-term improvements in data compression algorithms during R&D cycles.
Very distant and indirect implications for AI model efficiency or data communication, if ever practically applied at scale.
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Read at arXiv cs.AI