Approximate sparse linear regression
Name
LIPIcs-ICALP-2018-77.pdf
Description
Published version
Size
623.14 KB
Format
Adobe PDF
Checksum (MD5)
a3b12e8473ed2028b43f9e291c4e3ede
Author(s)
Har-Peled, S
Indyk, P
Mahabadi, S
Date Issued
July 1, 2018
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
Har-Peled, S., P. Indyk, and S. Mahabadi. "Approximate Sparse Linear Regression [Arxiv]." arXiv (2016): 19 pp.
Version
Final published version
Abstract
© 2018 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved. In the Sparse Linear Regression (SLR) problem, given a d × n matrix M and a d-dimensional query q, the goal is to compute a k-sparse n-dimensional vector τ such that the error Mτ − q is minimized. This problem is equivalent to the following geometric problem: given a set P of n points and a query point q in d dimensions, find the closest k-dimensional subspace to q, that is spanned by a subset of k points in P. In this paper, we present data-structures/algorithms and conditional lower bounds for several variants of this problem (such as finding the closest induced k dimensional flat/simplex instead of a subspace). In particular, we present approximation algorithms for the online variants of the above problems with query timeO(nk−1), which are of interest in the "low sparsity regime" where k is small, e.g., 2 or 3. For k = d, this matches, up to polylogarithmic factors, the lower bound that relies on the a nely degenerate conjecture (i.e., deciding if n points in Rd contains d+ 1 points contained in a hyperplane takes (nd) time). Moreover, our algorithms involve formulating and solving several geometric subproblems, which we believe to be of independent interest.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4230/LIPIcs.ICALP.2018.77
