A \emph{tensor-relational} computation is a relational computation where individual tuples carry vectors, matrices, or higher-dimensional arrays. An advantage of tensor-relational computation is that the overall computation can be executed on top of a relational system, inheriting the system's ability to automatically handle very large inputs with high levels of sparsity while high-performance kernels (such as optimized matrix-matrix multiplication codes) can be used to perform most of the underlying mathematical operations. In this paper, we introduce upper-case-lower-case \texttt{EinSum}, which is a tensor-relational version of the classical Einstein Summation Notation. We study how to automatically rewrite a computation in Einstein Notation into upper-case-lower-case \texttt{EinSum} so that computationally intensive components are executed using efficient numerical kernels, while sparsity is managed relationally.

翻译：张量-关系计算是一种关系计算，其中每个元组携带向量、矩阵或更高维数组。张量-关系计算的一个优势在于，整体计算可以在关系系统之上执行，从而继承系统自动处理具有高度稀疏性的超大规模输入的能力，同时可以使用高性能内核（如优化的矩阵-矩阵乘法代码）来执行大部分底层数学运算。本文介绍了大写-小写 \texttt{EinSum}，它是经典爱因斯坦求和记号的张量-关系版本。我们研究了如何将爱因斯坦记号中的计算自动重写为大写-小写 \texttt{EinSum}，从而使得计算密集型组件能够使用高效的数值内核执行，而稀疏性则通过关系方式管理。