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# Sparse Regression via Range Counting

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## Acknowledgements

The authors wish to thank the reviewers of earlier versions of this manuscript, who provided useful comments, as well as John Iacono and Stefan Langerman for insightful discussions.

## Cite As

Jean Cardinal and Aurélien Ooms. Sparse Regression via Range Counting. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.20

## Abstract

The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set S of n points in ℝ^d, a point y∈ ℝ^d, and an integer 2 ≤ k ≤ d, find an affine combination of at most k points of S that is nearest to y. We describe a O(n^{k-1} log^{d-k+2} n)-time randomized (1+ε)-approximation algorithm for this problem with d and ε constant. This is the first algorithm for this problem running in time o(n^k). Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP'18), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where k = d = O(1), we provide a simple O_δ(n^{d-1+δ})-time deterministic exact algorithm, for any δ > 0. Finally, we show how to adapt the approximation algorithm for the sparse linear regression and sparse convex regression problems with the same running time, up to polylogarithmic factors.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Computational geometry
• Information systems → Nearest-neighbor search
##### Keywords
• Sparse Linear Regression
• Orthogonal Range Searching
• Affine Degeneracy Testing
• Nearest Neighbors
• Hyperplane Arrangements

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