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

Tile low rank representations of dense matrices partition them into blocks of roughly uniform size, where each off-diagonal tile is compressed and stored as its own low rank factorization. They offer an attractive representation for many data-sparse dense operators that appear in practical applications, where substantial compression and a much smaller memory footprint can be achieved. TLR matrices are a compromise between the simplicity of a regular perfectly-strided data structure and the optimal complexity of the unbalanced trees of hierarchically low rank matrices, and provide a convenient performance-tuning parameter through their tile size that can be proportioned to take into account the cache size where the tiles reside in the memory hierarchy. There are currently no high-performance algorithms that can generate Cholesky and $LDL^T$ factorizations, particularly on GPUs. The difficulties in achieving high performance when factoring TLR matrices come from the expensive compression operations that must be performed during the factorization process and the adaptive rank distribution of the tiles that causes an irregular work pattern for the processing cores. In this work, we develop a dynamic batching operation and combine it with batched adaptive randomized approximations to achieve high performance both on GPUs and CPUs. Our implementation attains over 1.2 TFLOP/s in double precision on the V100 GPU, and is limited by the performance of batched GEMM operations. The Cholesky factorization of covariance matrix of size $N = 131K$ arising in spatial statistics can be factored to an accuracy $epsilon=10^{-2}$ in just a few seconds. We believe the proposed GEMM-centric algorithm allows it to be readily ported to newer hardware such as the tensor cores that are optimized for small GEMM operations.