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We present parallel algorithms and data structures for three fundamentaloperations in Numerical Linear Algebra: (i) Gaussian and CountSketch randomprojections and their combination, (ii) computation of the Gram matrix and(iii) computation of the squared row norms of the product of two matrices, witha special focus on "tall-and-skinny" matrices, which arise in manyapplications. We provide a detailed analysis of the ubiquitous CountSketchtransform and its combination with Gaussian random projections, accounting formemory requirements, computational complexity and workload balancing. We alsodemonstrate how these results can be applied to column subset selection, leastsquares regression and leverage scores computation. These tools have beenimplemented in pylspack, a publicly available Python package(https://github.com/IBM/pylspack) whose core is written in C++ and parallelizedwith OpenMP, and which is compatible with standard matrix data structures ofSciPy and NumPy. Extensive numerical experiments indicate that the proposedalgorithms scale well and significantly outperform existing libraries fortall-and-skinny matrices.

pylspack: Parallel algorithms and data structures for sketching, column subset selection, regression and leverage scores