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Efficient recursive least squares solver for rank-deficient matrices
arXiv - CS - Mathematical Software Pub Date : 2021-06-22 , DOI: arxiv-2106.11594
Ruben StaubLC, Stephan N. SteinmannLC

Updating a linear least squares solution can be critical for near real-time signalprocessing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A $\in$ R nxm with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least squares algorithm exploiting the rank-deficiency of A, achieving the update of the minimum-norm least-squares solution in O(mr) operations and, therefore, solving the linear least-squares problem from scratch in O(nmr) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity.

中文翻译:

秩亏矩阵的高效递归最小二乘求解器

更新线性最小二乘解对于近乎实时的信号处理应用至关重要。Greville 算法提出了一个简单的公式来更新秩为 r 的矩阵 A $\in$ R nxm 的伪逆。在本文中,我们通过保持一般秩分解明确推导出一个类似的公式,我们称之为 rank-Greville。基于这个公式,我们实现了一个递归最小二乘算法,利用了 A 的秩不足,实现了 O(mr) 操作中的最小范数最小二乘解的更新,因此,解决了线性最小二乘问题在 O(nmr) 操作中划伤。我们凭经验证实,对于秩亏矩阵,该算法显示出比 LAPACK 求解器更好的渐近时间复杂度。发现 rank-Greville 的数值稳定性与基于 Cholesky 的求解器相当。尽管如此,由于其显着的代数简单性,我们的实现支持有理数的精确数值表示。
更新日期:2021-06-25
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