Subspace Embedding and Linear Regression with Orlicz Norm
Alexandr Andoni, Chengyu Lin, Ying Sheng, Peilin Zhong, Ruiqi Zhong
Abstract
We consider a generalization of the classic linear regression problem to the case when the loss is an Orlicz norm. An Orlicz norm is parameterized by a non-negative convex function G: R_+ - > R_+ with G(0) = 0: the Orlicz norm of a n-dimensional vector x is defined as |x|_G = inf{ alpha > 0 | sum_{i = 1}^n G( |x_i| / alpha ) < = 1 }. We consider the cases where the function G grows subquadratically. Our main result is based on a new oblivious embedding which embeds the column space of a given nxd matrix A with Orlicz norm into a lower dimensional space with L2 norm. Specifically, we show how to efficiently find an mxn embedding matrix S (m < n), such that for every d-dimensional vector x, we have Omega(1/(d log n)) |Ax|_G < = |SAx|_2 < = O(d^2 log n) |Ax|_G. By applying this subspace embedding technique, we show an approximation algorithm for the regression problem min_x |Ax-b|_G, up to a O( d log^2 n ) factor. As a further application of our techniques, we show how to also use them to improve on the algorithm for the Lp low rank matrix approximation problem for 1 < = p < 2.
BibTeX
@InProceedings{pmlr-v80-andoni18a,
title = {Subspace Embedding and Linear Regression with Orlicz Norm},
author = {Andoni, Alexandr and Lin, Chengyu and Sheng, Ying and Zhong, Peilin and Zhong, Ruiqi},
booktitle = {Proceedings of the 35th International Conference on Machine Learning},
pages = {224--233},
year = {2018},
editor = {Dy, Jennifer and Krause, Andreas},
volume = {80},
series = {Proceedings of Machine Learning Research},
month = {10--15 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v80/andoni18a/andoni18a.pdf},
url = {https://proceedings.mlr.press/v80/andoni18a.html},
abstract = {We consider a generalization of the classic linear regression problem to the case when the loss is an Orlicz norm. An Orlicz norm is parameterized by a non-negative convex function G: R_+ - > R_+ with G(0) = 0: the Orlicz norm of a n-dimensional vector x is defined as |x|_G = inf{ alpha > 0 | sum_{i = 1}^n G( |x_i| / alpha ) < = 1 }. We consider the cases where the function G grows subquadratically. Our main result is based on a new oblivious embedding which embeds the column space of a given nxd matrix A with Orlicz norm into a lower dimensional space with L2 norm. Specifically, we show how to efficiently find an mxn embedding matrix S (m < n), such that for every d-dimensional vector x, we have Omega(1/(d log n)) |Ax|_G < = |SAx|_2 < = O(d^2 log n) |Ax|_G. By applying this subspace embedding technique, we show an approximation algorithm for the regression problem min_x |Ax-b|_G, up to a O( d log^2 n ) factor. As a further application of our techniques, we show how to also use them to improve on the algorithm for the Lp low rank matrix approximation problem for 1 < = p < 2.}
}