The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a real d \times n matrix A, a vector yin^d, a point y\in } R^d, and an integer 2 \leq k \leq d, find an affine combination of at most k columns of A that is closest to y. We describe a O(n^{k-1} \log^{d-k+2} n)-time randomized (1+\varepsilon)-approximation algorithm for this problem with d and \varepsilon 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 also provide a simple O (n ^{d-1+ \epsilon })-time deterministic exact algorithm  . 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.
The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set S of n points in \mathbb{R^d, a point y\in } R^d, and an integer 2 \leq k \leq 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+\varepsilon)-approximation algorithm for this problem with \(d\) and \(\varepsilon\) 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 also provide a simple O _\delta(n ^{d-1+ \delta })-time deterministic exact algorithm , for any \(\delta > 0\) . 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.
R
real d \times n matrix A, a vector yin
set S of n points in \mathbb{R
R
columns of A that is closest
points of S that is nearest
R
d and \varepsilon
\(d\) and \(\varepsilon\)
R
(n
_\delta(n
R
\epsilon
\delta
A
, for any \(\delta > 0\)