Approximate Sparse Linear Regression

Authors Sariel Har-Peled, Piotr Indyk, Sepideh Mahabadi



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Author Details

Sariel Har-Peled
  • Department of Computer Science, University of Illinois, Urbana, IL, USA
Piotr Indyk
  • Department of Computer Science, MIT, Cambridge, MA, USA
Sepideh Mahabadi
  • Data Science Institute, Columbia University, New York, NY, USA

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Sariel Har-Peled, Piotr Indyk, and Sepideh Mahabadi. Approximate Sparse Linear Regression. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.77

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Sparse Linear Regression
  • Approximate Nearest Neighbor
  • Sparse Recovery
  • Nearest Induced Flat
  • Nearest Subspace Search

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