Linear Regression is a seminal technique in statistics and machine learning,
where the objective is to build linear predictive models between a response
(i.e., dependent) variable and one or more predictor (i.e., independent)
variables. In this paper, we revisit the classical technique of Quantile
Regression (QR), which is statistically a more robust alternative to the other
classical technique of Ordinary Least Square Regression (OLS). However, while
there exist efficient algorithms for OLS, almost all of the known results for
QR are only weakly polynomial. Towards filling this gap, this paper proposes
several efficient strongly polynomial algorithms for QR for various settings.
For two dimensional QR, making a connection to the geometric concept of
$k$-set, we propose an algorithm with a deterministic worst-case time
complexity of $\mathcal{O}(n^{4/3} polylog(n))$ and an expected time complexity
of $\mathcal{O}(n^{4/3})$ for the randomized version. We also propose a
randomized divide-and-conquer algorithm -- RandomizedQR with an expected time
complexity of $\mathcal{O}(n\log^2{(n)})$ for two dimensional QR problem. For
the general case with more than two dimensions, our RandomizedQR algorithm has
an expected time complexity of $\mathcal{O}(n^{d-1}\log^2{(n)})$.

本文提出了针对 Quantile Regression 的多个高效确定性和随机多项式算法，连接到 k 集的几何概念和随机分治算法，分别解决了二维和多维 Quantile Regression 问题。