2026-08-15T12:17:02Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9081 2024-03-06T10:43:19Z ddc:004 open_access

Approximate Sparse Linear Regression Har-Peled, Sariel Indyk, Piotr Mahabadi, Sepideh Sparse Linear Regression Approximate Nearest Neighbor Sparse Recovery Nearest Induced Flat Nearest Subspace Search 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sariel Har-Peled and Piotr Indyk and Sepideh Mahabadi 2018 Is Part Of LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2018.77 urn:nbn:de:0030-drops-90816 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.77 eng https://creativecommons.org/licenses/by/3.0/legalcode