In this work, we explore the problem of multi-user linearly-separable distributed computation, where $N$ servers help compute the desired functions (jobs) of $K$ users, and where each desired function can be written as a linear combination of up to $L$ (generally non-linear) subtasks (or sub-functions). Each server computes some of the subtasks, communicates a function of its computed outputs to some of the users, and then each user collects its received data to recover its desired function. We explore the computation and communication relationship between how many servers compute each subtask vs. how much data each user receives. For a matrix $\mathbf{F}$ representing the linearly-separable form of the set of requested functions, our problem becomes equivalent to the open problem of sparse matrix factorization $\mathbf{F} = \mathbf{D}\mathbf{E}$ over finite fields, where a sparse decoding matrix $\mathbf{D}$ and encoding matrix $\mathbf{E}$ imply reduced communication and computation costs respectively. This paper establishes a novel relationship between our distributed computing problem, matrix factorization, syndrome decoding and covering codes. To reduce the computation cost, the above $\mathbf{D}$ is drawn from covering codes or from a here-introduced class of so-called `partial covering' codes, whose study here yields computation cost results that we present.

翻译：在这项工作中,我们探索了多用户线性分布计算的问题, 在那里, $N服务器帮助计算每个子任务( job) 想要的功能( job) $K$ 用户, 并且每个想要的函数可以写成最多为 $( 一般来说非线性) 的线性组合 子任务( 或子函数 ) 。 每个服务器都计算了一些子任务( 或子函数 ), 向一些用户传达其计算产出的函数, 然后每个用户收集其收到的数据以恢复其想要的功能。 我们探索了计算和通讯关系之间的关系, 有多少服务器计算每个子任务( 工作) 想要计算( 工作) $K$( 工作) 和每个用户收到多少数据。 对于代表所请求的功能组的线性形式的矩阵 $\ mathbf{ F} 来说, 我们的问题就相当于空矩阵因子要素 $\ mathb f{ F} 的开放问题, 向一些用户传递一个函数函数的函数函数的函数函数的函数的函数函数, = = = 计算成本。