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Tile low rank representations of dense matrices partition them into blocks ofroughly uniform size, where each off-diagonal tile is compressed and stored asits own low rank factorization. They offer an attractive representation formany data-sparse dense operators that appear in practical applications, wheresubstantial compression and a much smaller memory footprint can be achieved.TLR matrices are a compromise between the simplicity of a regularperfectly-strided data structure and the optimal complexity of the unbalancedtrees of hierarchically low rank matrices, and provide a convenientperformance-tuning parameter through their tile size that can be proportionedto take into account the cache size where the tiles reside in the memoryhierarchy.  There are currently no high-performance algorithms that can generate Choleskyand $LDL^T$ factorizations, particularly on GPUs. The difficulties in achievinghigh performance when factoring TLR matrices come from the expensivecompression operations that must be performed during the factorization processand the adaptive rank distribution of the tiles that causes an irregular workpattern for the processing cores. In this work, we develop a dynamic batchingoperation and combine it with batched adaptive randomized approximations toachieve high performance both on GPUs and CPUs.  Our implementation attains over 1.2 TFLOP/s in double precision on the V100GPU, and is limited by the performance of batched GEMM operations. The Choleskyfactorization of covariance matrix of size $N = 131K$ arising in spatialstatistics can be factored to an accuracy $\epsilon=10^{-2}$ in just a fewseconds. We believe the proposed GEMM-centric algorithm allows it to be readilyported to newer hardware such as the tensor cores that are optimized for smallGEMM operations.

H2OPUS-TLR: High Performance Tile Low Rank Symmetric Factorizations using Adaptive Randomized Approximation