An Upgrading Algorithm With Optimal Power Law

Abstract

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. We complement our analysis with numerical experiments which show that the developed algorithm improves upon the existing state-of-the-art algorithms also in non-asymptotic setups.