The Sphere Packing Bound For Memoryless Channels

Sphere packing bounds (SPBs) ---with prefactors that are polynomial in the block length--- are derived for codes on two families of memoryless channels using Augustin's method: (possibly non-stationary) memoryless channels with (possibly multiple) additive cost constraints and stationary memoryless channels with convex constraints on the composition (i.e. empirical distribution, type) of the input codewords. A variant of Gallager's bound is derived in order to show that these sphere packing bounds are tight in terms of the exponential decay rate of the error probability with the block length under mild hypotheses.Comment: 29 page

The Augustin Capacity and Center

For any channel, the existence of a unique Augustin mean is established for any positive order and probability mass function on the input set. The Augustin mean is shown to be the unique fixed point of an operator defined in terms of the order and the input distribution. The Augustin information is shown to be continuously differentiable in the order. For any channel and convex constraint set with finite Augustin capacity, the existence of a unique Augustin center and the associated van Erven-Harremoes bound are established. The Augustin-Legendre (A-L) information, capacity, center, and radius are introduced and the latter three are proved to be equal to the corresponding Renyi-Gallager quantities. The equality of the A-L capacity to the A-L radius for arbitrary channels and the existence of a unique A-L center for channels with finite A-L capacity are established. For all interior points of the feasible set of cost constraints, the cost constrained Augustin capacity and center are expressed in terms of the A-L capacity and center. Certain shift invariant families of probabilities and certain Gaussian channels are analyzed as examples.Comment: 59 page

A derivation of the cost-constrained sphere-packing exponent

We derive the channel-coding sphere-packing exponent under a per-codeword cost constraint. The proof is based on hypothesis testing and holds for continuous memoryless channels

Divergence Measures

Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures