Subspace Embedding and Linear Regression with Orlicz Norm

Alexandr Andoni, Chengyu Lin, Ying Sheng, Peilin Zhong, Ruiqi Zhong
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:224-233, 2018.

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.

Cite this Paper


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 = {http://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.} }
Endnote
%0 Conference Paper %T Subspace Embedding and Linear Regression with Orlicz Norm %A Alexandr Andoni %A Chengyu Lin %A Ying Sheng %A Peilin Zhong %A Ruiqi Zhong %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-andoni18a %I PMLR %P 224--233 %U http://proceedings.mlr.press/v80/andoni18a.html %V 80 %X 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.
APA
Andoni, A., Lin, C., Sheng, Y., Zhong, P. & Zhong, R.. (2018). Subspace Embedding and Linear Regression with Orlicz Norm. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:224-233 Available from http://proceedings.mlr.press/v80/andoni18a.html.

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