Multiplicative Perturbation Bounds for Multivariate Multiple Linear Regression in Schatten p-Norms [article]

Jocelyn T. Chi, Ilse C. F. Ipsen
2020 arXiv   pre-print
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
more » ... rs 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.
arXiv:2007.06099v1 fatcat:vea5vuujrnc3ngjop33b53mcju