Cholesky分解的微分
Differentiation of the Cholesky decomposition
摘要：
我们审查战略，区分矩阵为基础的计算，并推导出符号和算法的更新规则，区分表达式包含的Cholesky分解。我们推荐新的“阻塞”算法，基于区分LAPACK库中的Cholesky算法DPOTRF，它使用BLAS的“第3级”矩阵-矩阵运算，因此是高速缓存友好的，易于并行化。对于大型矩阵，所得到的算法是计算Cholesky导数的最快方法，并且比常用的算法快一个数量级。在某些计算环境中，对于小矩阵，符号导出的更新比基于差分Cholesky算法的更新更快。符号和算法方法可以结合起来，以获得最好的两个世界。
英文摘要：
We review strategies for differentiating matrix-based computations, and derive symbolic and algorithmic update rules for differentiating expressions containing the Cholesky decomposition. We recommend new `blocked' algorithms, based on differentiating the Cholesky algorithm DPOTRF in the LAPACK library, which uses `Level 3' matrix-matrix operations from BLAS, and so is cache-friendly and easy to parallelize. For large matrices, the resulting algorithms are the fastest way to compute Cholesky derivatives, and are an order of magnitude faster than the algorithms in common usage. In some computing environments, symbolically-derived updates are faster for small matrices than those based on differentiating Cholesky algorithms. The symbolic and algorithmic approaches can be combined to get the best of both worlds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
