A Family of Algorithms for Computing Information-Theoretic Forms of Strong Converse Exponents in Channel Coding and Lossy Source Coding

This paper studies the computation of error and correct decoding probability exponents in channel coding and lossy source coding. For channel coding, we show that the recently proposed algorithm of Tridenski and Zamir for computing strong converse exponent has the attractive property that the objective function alternately takes the forms appearing in Arimoto's and Dueck and K\"orner's exponents. Because of this property, the convergence of the algorithm directly implies the match of the two exponents. Then, we consider a special case of Tridenski et al.'s generalized algorithm of that has not been examined in depth. We show that the objective function of this algorithm also has the interesting property that it takes the forms of Dueck and K\"orner's exponent and another representation of the strong converse exponent derived by Arimoto's algorithm. This particular case is important because the objective function in this case can be used to prove that the Gallager and Csisz\'ar and K\"orner error exponents agree.63 For lossy source coding, we propose two new algorithms for computing the Csisz\'ar and K\"orner's strong converse exponent. We also define a function similar to Blahut's error exponent with a negative slope parameter for source coding. We show that one of the proposed algorithms has the property that the objective function alternately takes the forms of Csisz\'ar and K\"orner's exponent and the newly defined exponent function. The convergence of the algorithm proves that the new exponent function coincides with the Csisz\'ar and K\"orner exponent. We also prove that Arimoto algorithm for computing error exponent of lossy source coding can work for negative $\rho \ge -1$ and thus can be used to compute the new exponent function. Computation of Arikan and Merhav's guessing exponent is also discussed.