Towards Structured Algebraic Programming

Structured matrices and tensors exhibiting properties such as symmetry and fixed non-zero patterns are known for making algorithms and data storage more efficient. Due to emerging power and efficiency constraints required by the scale of modern scientific, machine learning, and edge computing applications, algorithm experts are revisiting traditional linear algebra algorithms with the goal of making new structures appear. Such structures often result from new numerical approximations that would greatly benefit from a more flexible linear algebra interface than standard BLAS and LAPACK, allowing for mixed precision and data types to appear in place of traditional floating-point operations. Algebraic programming interfaces, like GraphBLAS, while naturally abstracting the algebras of their operations, do not explicitly capture structured, densely stored arrays. In this paper, we present a preliminary design of a new algebraic programming interface for structured containers with template-generic, non-zero patterns. This interface offers to backend implementations the possibility of integrating more compile-time pattern information in the loop-based implementation of primitive operations as well as in the formulation of array accesses. We demonstrate its ability to specify important dense matrix decomposition algorithms and argue its ability to expose high-performance backends.