设计可精确控制输出分布的最优量化方法，降低重构误差。
Minimum Distortion Quantization with Specified Output Distribution
- 基于指定输出分布构造最小均方误差量化器
- 当输入或输出为均匀分布时，量化器形式简化为累积分布函数映射
- 适用于通信、数据匿名化等需精准控制输出熵的场景
我们推导了实值随机变量 $W$（分布为 $P_W$）的最优量化器，使得量化输出 $X$ 取 $k$ 个值且服从指定分布 $P_X$ 于 $\\(1,\ldots,k\\)，同时最小化从 $X$ 估计 $W$ 的最小均方误差（MMSE）。最优量化器形式为 $X=σ\big(F_{σ^{-1}(X)}^{-1}(F_W(W))\big)$，其中 $σ$ 是使 MMSE 最小的 $\\{1,\ldots,k\ }$ 上的最优排列。当 $P_W$ 在区间上均匀或 $P_X$ 均匀时，量化器简化为 $X=F_{X}^{-1}(F_W(W))$。主要性在最优性证明中起关键作用。指定输出分布可用于设计具有明确输出熵、最大化输入输出互信息、匹配信道输入需求或用于数据匿名化的量化器。
原文摘要 · Abstract (English)
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 $X=F_{X}^{-1}(F_W(W))$. The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information between input and output, tailored output distribution to match channel input requirements for communication, and data anonymization.
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