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Linear algebra operations have been widely used in big data analytics andscientific computations. Many works have been done on optimizing linear algebraoperations on GPUs with regular-shaped input. However, few works focus on fullyutilizing GPU resources when the input is not regular-shaped. Currentoptimizations do not consider fully utilizing the memory bandwidth andcomputing power; therefore, they can only achieve sub-optimal performance. Inthis paper, we propose two efficient algorithms -- TSM2R and TSM2L -- for twoclasses of tall-and-skinny matrix-matrix multiplications on GPUs. Both of themfocus on optimizing linear algebra operation with at least one of the inputmatrices is tall-and-skinny. Specifically, TSM2R is designed for a largeregular-shaped matrix multiplying a tall-and-skinny matrix, while TSM2L isdesigned for a tall-and-skinny matrix multiplying a small regular-shapedmatrix. We implement our proposed algorithms and test on several modern NVIDIAGPU micro-architectures. Experiments show that, compared to the currentstate-of-the-art works, (1) TSM2R speeds up the computation by 1.1x~3x andimproves the memory bandwidth utilization and computing power utilization by8%~47.6% and 7%~37.3%, respectively, when the regular-shaped matrix size isrelatively large or medium; and (2) TSM2L speeds up the computation by1.1x~3.5x and improve the memory bandwidth utilization by up to 55% when theregular-shaped matrix size is relatively small.

TSM2X: High-Performance Tall-and-Skinny Matrix-Matrix Multiplication on GPUs