Multiplicative perturbation bounds for multivariate multiple linear regression in Schatten p-norms
Multiplicative perturbation bounds for multivariate multiple linear regression in Schatten p-norms
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DOI：
10.1016/j.laa.2021.03.039
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发表时间：
2021-04-19
影响因子：
1.1
通讯作者：
Ipsen, Ilse C. F.
中科院分区：
文献类型：
--
作者：
Chi, Jocelyn T.;Ipsen, Ilse C. F.
多元多元线性回归(MMLR)将传统的最小二乘(多元线性回归)推广到多个右端，在实际应用中得到了广泛的应用。通过将素描问题解释为乘性扰动，我们将最近的MLR分析推广到一般Schatten p-范数下的素描MMLR。我们的工作是Maher关于Schatten p-范数结果的推广。为了便于几何解释，我们用投影法推导出精确解和摄动解的表达式。我们还给出了草图矩阵在相应子空间中作用的几何解释。我们证明了在某些假设下，评价MMLR解的精度的一个关键项可以看作是子空间之间最大主角的切线。我们的结果可以进一步解释具有相同射程的正交和倾斜投影仪之间的差异。(C)2021 Elsevier Inc.保留所有权利。
Multivariate multiple linear regression (MMLR), which occurs in a number of practical applications, generalizes traditional least squares (multivariate linear regression) to multiple right-hand sides. We extend recent MLR analyses to sketched MMLR in general Schatten p-norms by interpreting the sketched problem as a multiplicative perturbation. Our work represents an extension of Maher's results on Schatten p-norms. We derive expressions for the exact and perturbed solutions in terms of projectors for easy geometric interpretation. We also present a geometric interpretation of the action of the sketching matrix in terms of relevant subspaces. We show that a key term in assessing the accuracy of the sketched MMLR solution can be viewed as a tangent of a largest principal angle between subspaces under some assumptions. Our results enable additional interpretation of the difference between an orthogonal and oblique projector with the same range. (C) 2021 Elsevier Inc. All rights reserved.