本文提出 rpCholesky-QR 随机预条件 Cholesky-QR 算法，用于满秩矩阵薄 QR 分解，证明其低误差、稳定不崩溃及二范数扰动界，并通过实验验证其精度和条件数依赖优势。
摘要：
我们提出并分析了 rpCholesky-QR，这是一种随机预条件 Cholesky-QR 算法，用于计算秩为 n 的实数 m×n 矩阵的薄 QR 分解。rpCholesky-QR 具有较低的正交化误差、机器精度量级的残差，并且对于高度奇异的矩阵也不会发生崩溃。我们在最少假设下，为 rpCholesky-QR 推导了严格且可解释的二范数扰动界。数值实验证实了仅从少至 3n 行采样预条件子时 rpCholesky-QR 的准确性，并说明偏离正交规范性的二范数偏差仅随预条件后矩阵的条件数而非其平方增长——即使原始矩阵在数值上是奇异的。
英文摘要：
We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of machine precision, and does not break down for highly singular matrices. We derive rigorous and interpretable two-norm perturbation bounds for rpCholesky-QR that require a minimum of assumptions. Numerical experiments corroborate the accuracy of rpCholesky-QR for preconditioners sampled from as few as 3n rows, and illustrate that the two-norm deviation from orthonormality increases with only the condition number of the preconditioned matrix, rather than its square -- even if the original matrix is numerically singular.
CholeskyQR is a simple and fast QR decomposition via Cholesky decomposition, while it has been considered highly sensitive to the condition number. In this paper, we provide a randomized preconditioner framework for CholeskyQR algorithm. Under this framework, two methods (randomized LU-CholeskyQR and randomized QR-CholeskyQR) are proposed and discussed. We prove the proposed preconditioners can effectively reduce the condition number, which is also demonstrated by numerical tests. Abundant numerical tests indicate our methods are more stable and faster than all the existing algorithms and have…
This article proposes and analyzes several variants of the randomized Cholesky QR factorization of a matrix $X$. Instead of computing the R factor from $X^T X$, as is done by standard methods, we obtain it from a small, efficiently computable random sketch of $X$, thus saving computational cost and improving numerical stability. The proposed direct variant of the randomized Cholesky QR requires only half the flops and the same communication cost as the classical Cholesky QR. At the same time, it is more robust since it is guaranteed to be stable whenever the input matrix is numerically full-ra…
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