Partial Information Decomposition via Deficiency for Multivariate Gaussians

Bivariate partial information decompositions (PIDs) characterize how the information in a "message" random variable is decomposed between two "constituent" random variables in terms of unique, redundant and synergistic information components. These components are a function of the joint distribution of the three variables, and are typically defined using an optimization over the space of all possible joint distributions. This makes it computationally challenging to compute PIDs in practice and restricts their use to low-dimensional random vectors. To ease this burden, we consider the case of jointly Gaussian random vectors in this paper. This case was previously examined by Barrett (2015), who showed that certain operationally well-motivated PIDs reduce to a closed form expression for scalar messages. Here, we show that Barrett's result does not extend to vector messages in general, and characterize the set of multivariate Gaussian distributions that reduce to closed-form. Then, for all other multivariate Gaussian distributions, we propose a convex optimization framework for approximately computing a specific PID definition based on the statistical concept of deficiency. Using simplifying assumptions specific to the Gaussian case, we provide an efficient algorithm to approximately compute the bivariate PID for multivariate Gaussian variables with tens or even hundreds of dimensions. We also theoretically and empirically justify the goodness of this approximation.Comment: Presented at ISIT 2022. This version has been updated to reflect the final conference publication, including appendices. It also corrects technical errors in Remark 1 and Appendix C, adds a new experiment, and has a substantially improved presentation as well as additional detail in the appendix, compared to the previous arxiv versio

Capacity Region of Vector Gaussian Interference Channels with Generally Strong Interference

An interference channel is said to have strong interference if for all input distributions, the receivers can fully decode the interference. This definition of strong interference applies to discrete memoryless, scalar and vector Gaussian interference channels. However, there exist vector Gaussian interference channels that may not satisfy the strong interference condition but for which the capacity can still be achieved by jointly decoding the signal and the interference. This kind of interference is called generally strong interference. Sufficient conditions for a vector Gaussian interference channel to have generally strong interference are derived. The sum-rate capacity and the boundary points of the capacity region are also determined.Comment: 50 pages, 11 figures, submitted to IEEE trans. on Information Theor

Noisy-Interference Sum-Rate Capacity for Vector Gaussian Interference Channels

New sufficient conditions for a vector Gaussian interference channel to achieve the sum-rate capacity by treating interference as noise are derived, which generalize and extend the existing results. More concise conditions for multiple-input single-output, and single-input multiple-output scenarios are obtained

Noisy-Interference Sum-Rate Capacity for Vector Gaussian Interference Channels

New sufficient conditions for a vector Gaussian interference channel to achieve the sum-rate capacity by treating interference as noise are derived, which generalize and extend the existing results. More concise conditions for multiple-input single-output, and single-input multiple-output scenarios are obtained