The Capacity-Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points.

We study the real, complex, and vector additive Gaussian channels with input amplitude constraints. For the real additive Gaussian channel model with a peak power constraint on the input, a classical result by Smith implies that the capacity-achieving input distribution is discrete with finitely many mass points. Similarly, for the complex and vector additive Gaussian channels with input amplitude constraints, one can show that the multi-dimensional capacity achieving distributions in these respective settings will have shelled structures with finitely many shells. However, an unfortunate deficiency in these surprising and useful results is that Smith’s method (and its sibling methods in other similar problems) uses the “proof by contradiction” technique. While it is already too hard to find the capacity-achieving distributions in these problems, because of this contradiction-based proof technique, not even a bound on their support sizes was previously available. In this work, we provide an alternative and far simpler proof for the discrete nature of the capacity-achieving input distributions in these problems, and being a constructive technique as it is, this new method we show is able to produce an implicit upper bound on the number of mass points (or ‘number of shells’  in the complex and vector cases). This paves an alternative way in approaching many such problems in the literature.

Biography:
Semih received his Bachelor of Science degree in Electrical and Electronics Engineering in 2013, his Bachelor of Science degree in Mathematics in 2014 both from Middle East Technical University, and his Master of Arts degree in Electrical Engineering in 2016 from Princeton University. Currently, he is pursuing his Ph.D. degree in Electrical Engineering in Princeton University under the supervision of H. Vincent Poor. His research interest revolve around information theory, statistical modeling, optimization, theory of detection, estimation, and most recently, federated learning algorithms.