HarPeled, Sariel ;
Indyk, Piotr ;
Mahabadi, Sepideh
Approximate Sparse Linear Regression
Abstract
In the Sparse Linear Regression (SLR) problem, given a d x n matrix M and a ddimensional query q, the goal is to compute a ksparse ndimensional vector tau such that the error M tau  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 kdimensional subspace to q, that is spanned by a subset of k points in P. In this paper, we present datastructures/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 time O~(n^{k1}), 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 affinely degenerate conjecture (i.e., deciding if n points in R^d contains d+1 points contained in a hyperplane takes Omega(n^d) time). Moreover, our algorithms involve formulating and solving several geometric subproblems, which we believe to be of independent interest.
BibTeX  Entry
@InProceedings{harpeled_et_al:LIPIcs:2018:9081,
author = {Sariel HarPeled and Piotr Indyk and Sepideh Mahabadi},
title = {{Approximate Sparse Linear Regression}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {77:177:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9081},
URN = {urn:nbn:de:0030drops90816},
doi = {10.4230/LIPIcs.ICALP.2018.77},
annote = {Keywords: Sparse Linear Regression, Approximate Nearest Neighbor, Sparse Recovery, Nearest Induced Flat, Nearest Subspace Search}
}
2018
Keywords: 

Sparse Linear Regression, Approximate Nearest Neighbor, Sparse Recovery, Nearest Induced Flat, Nearest Subspace Search 
Seminar: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Issue date: 

2018 
Date of publication: 

2018 