Multi-dimensional Homomorphisms and Their Implementation in OpenCL

Homomorphisms (traditionally defined on lists) are functions that can be parallelized by the divide-and-conquer paradigm. In this paper, we introduce an extension of the traditional homomorphism concept—multi-dimensional homomorphisms (MDHs)—which capture parallelism on multi-dimensional arrays. We propose md_hom—a new parallel pattern (a.k.a. algorithmic skeleton), based on the MDH concept, to simplify parallel programming for a broad class of applications. The md_hom pattern is general enough to subsume common parallel patterns such as map and reduce, and also more complex functions built by composing and nesting several patterns. We present a generic implementation schema for md_hom in form of an efficient, correct-by-construction OpenCL pseudocode that targets various parallel architectures such as multi-core CPU and graphics processing unit (GPU). We develop our pseudocode schema as parametrized in tuning parameters: these allow to optimize the code for different devices and input sizes by performing an automated search on the parameter space. We evaluate the schematically generated, executable OpenCL code using the example of general matrix–vector multiplication (GEMV)—an important linear algebra routine which has gained more attention recently due to its use in the application area of deep learning—on two parallel architectures—Intel CPU and NVIDIA GPU. Our performance results are competitive and in some cases even better than the hand-tuned GEMV implementations provided by the state-of-the-art libraries Intel MKL and NVIDIA cuBLAS, as well as the auto-tunable OpenCL BLAS library CLBlast.