Distributed Source Coding for Compressing Vector-Linear Functions

Inspired by mobile satellite communication systems and the important and prevalent applications of computational tasks, we consider a distributed source coding model for compressing vector-linear functions, which consists of multiple sources, multiple encoders and a decoder linked to all the encoders. Each encoder has access to a certain subset of the sources and the decoder is required to compute with zero error a vector-linear function of the source information, which corresponds to a matrix $T$. The connectivity state between the sources and the encoders and the vector-linear function are all arbitrary. In the paper, we are interested in the function-compression capacity to measure the efficiency of using the system. We first present a general lower bound on the function-compression capacity applicable to arbitrary connectivity states and vector-linear functions. Next, we confine to the nontrivial models with only three sources and no more than three encoders, and prove that all the $3\times2$ column-full-rank matrices $T$ can be divided into two types $T_1$ and $T_2$, for which the function-compression capacities are identical if the matrices $T$ have the same type. We explicitly characterize the function-compression capacities for two most nontrivial models associated with $T_2$ by a novel approach of both upper bounding and lower bounding the size of image sets of encoding functions. This shows that the lower bound thus obtained is not always tight. Rather, by completely characterizing their capacities, the lower bound is tight for all the models associated with $T_1$ and all the models associated with $T_2$ except for the two most nontrivial models. We finally apply the obtained results to network function computation and answer the open problem whether the best known upper bound proved by Guang et. al. (2019) on computing capacity is in general asymptotically tight.