多元多元线性回归在Schatten $p$ -范数下的乘性扰动界
Multiplicative Perturbation Bounds for Multivariate Multiple Linear Regression in Schatten $p$-Norms
摘要：
多元多元线性回归（MMLR）是将传统的最小二乘法（多元线性回归）推广到多个右端的方法，在实际应用中得到了广泛的应用。我们扩展最近的MLR分析草图MMLR一般Schatten $p$ -规范的解释作为一个乘法扰动的草图问题。我们的工作是Maher关于Schatten $p$ -范数结果的推广。我们推导出表达式的精确和扰动的解决方案的投影仪，便于几何解释。我们还提出了一个几何解释的草图矩阵的行动有关的子空间。我们表明，在评估的准确性的草图MMLR解决方案的一个关键词可以被视为一个最大的主角之间的子空间在某些假设下的切线。我们的研究结果使额外的解释之间的差异，正交和倾斜投影仪具有相同的范围。
英文摘要：
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.
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