Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Har-Peled, Sariel; Indyk, Piotr; Mahabadi, Sepideh http://www.dagstuhl.de/lipics License
when quoting this document, please refer to the following
DOI:
URN: urn:nbn:de:0030-drops-90816
URL:

; ;

Approximate Sparse Linear Regression

pdf-format:


Abstract

In the Sparse Linear Regression (SLR) problem, given a d x n matrix M and a d-dimensional query q, the goal is to compute a k-sparse n-dimensional 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 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 time O~(n^{k-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 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 Har-Peled and Piotr Indyk and Sepideh Mahabadi},
  title =	{{Approximate Sparse Linear Regression}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9081},
  URN =		{urn:nbn:de:0030-drops-90816},
  doi =		{10.4230/LIPIcs.ICALP.2018.77},
  annote =	{Keywords: Sparse Linear Regression, Approximate Nearest Neighbor, Sparse Recovery, Nearest Induced Flat, Nearest Subspace Search}
}

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


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI