Consider a channel $W$ along with a given input distribution $P_X$. In certain settings, such as in the construction of polar codes, the output alphabet of $W$ is `too large', and hence we replace $W$ by a channel $Q$ having a smaller output alphabet. We say that $Q$ is upgraded with respect to $W$ if $W$ is obtained from $Q$ by processing its output. In this case, the mutual information $I(P_X,W)$ between the input and output of $W$ is upper-bounded by the mutual information $I(P_X,Q)$ between the input and output of $Q$. In this paper, we present an algorithm that produces an upgraded channel $Q$ from $W$, as a function of $P_X$ and the required output alphabet size of $Q$, denoted $L$. We show that the difference in mutual informations is `small'. Namely, it is $O(L^{-2/(|\mathcal{X}|-1)})$, where $|\mathcal{X}|$ is the size of the input alphabet. This power law of $L$ is optimal.

中文翻译：

---

具有最优幂律的升级算法

考虑通道 $W$ 以及给定的输入分布 $P_X$。在某些设置中，例如在极坐标代码的构建中，$W$ 的输出字母表“太大”，因此我们将 $W$ 替换为具有较小输出字母表的通道 $Q$。如果通过处理其输出从 $Q$ 获得 $W$，我们说 $Q$ 相对于 $W$ 升级。在这种情况下，$W$ 的输入和输出之间的互信息$I(P_X,W)$ 的上限是$Q$ 的输入和输出之间的互信息$I(P_X,Q)$。在本文中，我们提出了一种算法，该算法从 $W$ 生成升级通道 $Q$，作为 $P_X$ 的函数和所需的输出字母表大小 $Q$，表示为 $L$。我们表明互信息的差异是“小”的。即$O(L^{-2/(|\mathcal{X}|-1)})$，其中 $|\mathcal{X}|$ 是输入字母表的大小。这个 $L$ 的幂律是最优的。