We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen’s algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds by use of the rank expansion, or the minimum rank of any fixed size subset of columns of a matrix, for each of the three matrices encoding a bilinear algorithm. This framework provides lower bounds for a class of dependency directed acyclic graphs (DAGs) corresponding to the execution of a given bilinear algorithm, in contrast to other approaches that yield bounds for specific DAGs. However, our lower bounds only apply to executions that do not compute the same DAG node multiple times. Two bilinear algorithms can be nested by taking Kronecker products between their encoding matrices. Our main result is a lower bound on the rank expansion of a matrix constructed by a Kronecker product derived from lower bounds on the rank expansion of the Kronecker product’s operands. We apply the rank expansion lower bounds to obtain novel communication lower bounds for nested Toom-Cook convolution, Strassen’s algorithm, and fast algorithms for contraction of partially symmetric tensors.

我们为嵌套双线性算法（例如 Strassen 的矩阵乘法算法）的内存层次结构或处理器之间的通信制定了下限。我们建立在以前的框架的基础上，该框架通过使用秩扩展或矩阵列的任何固定大小子集的最小秩来建立通信下界，对于编码双线性算法的三个矩阵中的每一个。与为特定 DAG 生成边界的其他方法相比，该框架为与给定双线性算法的执行相对应的一类依赖有向无环图 (DAG) 提供了下限。但是，我们的下限仅适用于不多次计算同一 DAG 节点的执行。两个双线性算法可以通过在其编码矩阵之间采用克罗内克积来嵌套。我们的主要结果是由克罗内克积构造的矩阵的秩展开的下界，该矩阵是从克罗内克积的操作数的秩展开的下界导出的。我们应用秩扩展下界来获得嵌套 Toom-Cook 卷积、Strassen 算法和部分对称张量收缩的快速算法的新颖通信下界。