2020/04/02 by Or Ordentlich, Ordentlich, Or, Ido Tal +1
Computer Science · Mathematics · #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.2004.00869
arxiv created 2021/05/28 · arxiv updated 2021/05/31
Consider a channel W along with a given input distribution PX. 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(PX,W) between the input and output of W is upper-bounded by the mutual information I(PX,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 PX 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/(|X|-1)), where |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.