The work considers the $N$-server distributed computing scenario with $K$ users requesting functions that are linearly-decomposable over an arbitrary basis of $L$ real (potentially non-linear) subfunctions. In our problem, the aim is for each user to receive their function outputs, allowing for reduced reconstruction error (distortion) $\epsilon$, reduced computing cost ($\gamma$; the fraction of subfunctions each server must compute), and reduced communication cost ($\delta$; the fraction of users each server must connect to). For any given set of $K$ requested functions -- which is here represented by a coefficient matrix $\mathbf {F} \in \mathbb{R}^{K \times L}$ -- our problem is made equivalent to the open problem of sparse matrix factorization that seeks -- for a given parameter $T$, representing the number of shots for each server -- to minimize the reconstruction distortion $\frac{1}{KL}\|\mathbf {F} - \mathbf{D}\mathbf{E}\|^2_{F}$ overall $\delta$-sparse and $\gamma$-sparse matrices $\mathbf{D}\in \mathbb{R}^{K \times NT}$ and $\mathbf{E} \in \mathbb{R}^{NT \times L}$. With these matrices respectively defining which servers compute each subfunction, and which users connect to each server, we here design our $\mathbf{D},\mathbf{E}$ by designing tessellated-based and SVD-based fixed support matrix factorization methods that first split $\mathbf{F}$ into properly sized and carefully positioned submatrices, which we then approximate and then decompose into properly designed submatrices of $\mathbf{D}$ and $\mathbf{E}$.

翻译：摘要：本研究考虑一个$N$服务器分布式计算场景，其中$K$个用户请求的函数可基于$L$个实值（可能为非线性）子函数的任意基进行线性分解。目标是为每个用户提供其函数输出，同时降低重构误差（失真度）$\epsilon$、计算成本（$\gamma$；每个服务器需计算的子函数比例）和通信成本（$\delta$；每个服务器需连接的用户比例）。对于任意给定的$K$个请求函数（此处由系数矩阵$\mathbf {F} \in \mathbb{R}^{K \times L}$表示），该问题等价于稀疏矩阵分解这一开放性问题——即给定参数$T$（表示每个服务器的计算轮次），在全体$\delta$-稀疏与$\gamma$-稀疏矩阵$\mathbf{D}\in \mathbb{R}^{K \times NT}$和$\mathbf{E} \in \mathbb{R}^{NT \times L}$上，最小化重构失真度$\frac{1}{KL}\|\mathbf {F} - \mathbf{D}\mathbf{E}\|^2_{F}$。基于这些矩阵分别定义各服务器计算的子函数以及各用户连接的服务器，我们设计了一种基于嵌片（tessellation）和SVD的固定支撑矩阵分解方法，通过将$\mathbf{F}$拆分为尺寸适当且位置精心编排的子矩阵，随后对这些子矩阵进行近似并分解为$\mathbf{D}$和$\mathbf{E}$的规范子矩阵，从而构造$\mathbf{D}$和$\mathbf{E}$。