Communication Lower Bounds of Bilinear Algorithms for Symmetric Tensor Contractions
Provides theoretical insights into communication costs for symmetric tensor contractions, relevant to computational chemistry and data analysis, but the results are theoretical and incremental.
The paper introduces a theoretical framework for deriving communication lower bounds in bilinear algorithms, applied to symmetric tensor contractions. It shows that tensor symmetry, while reducing arithmetic cost and memory, can increase data movement by factors scaling with tensor size.
We introduce a new theoretical framework for deriving lower bounds on data movement in bilinear algorithms. Bilinear algorithms are a general representation of fast algorithms for bilinear functions, which include computation of matrix multiplication, convolution, and symmetric tensor contractions. A bilinear algorithm is described by three matrices. Our communication lower bounds are based on quantifying the minimal matrix ranks of matching subsets of columns of these matrices. This infrastructure yields new communication lower bounds for symmetric tensor contraction algorithms, which provide qualitative new insights. Tensor symmetry (invariance under permutation of modes) is common to many applications of tensor computations (e.g., tensor representation of hypergraphs, analysis of high order moments in data, as well as tensors modelling interactions of electrons in computational chemistry). Tensor symmetry enables reduction in representation size as well as arithmetic cost of contractions by factors that scale with the number of equivalent permutations. However, we derive lower bounds showing that these arithmetic cost and memory reductions can necessitate increases in data movement by factors that scale with the size of the tensors.