Fast and Layout-Oblivious Tensor-Matrix Multiplication with BLAS

The tensor-matrix multiplication is a basic tensor operation required by various tensor methods such as the ALS and the HOSVD. This paper presents flexible high-performance algorithms that compute the tensor-matrix product according to the Loops-over-GEMM (LoG) approach. Our algorithms can process dense tensors with any linear tensor layout, arbitrary tensor order and dimensions all of which can be runtime variable. We discuss different tensor slicing methods with parallelization strategies and propose six algorithm versions that call BLAS with subtensors or tensor slices. Their performance is quantified on a set of tensors with various shapes and tensor orders. Our best performing version attains a median performance of 1.37 double precision Tflops on an Intel Xeon Gold 6248R processor using Intel’s MKL. We show that the tensor layout does not affect the performance significantly. Our fastest implementation is on average at least 14.05% and up to 3.79x faster than other state-of-the-art approaches and actively developed libraries like Libtorch and Eigen.