Revisiting Quantum Algorithms for Linear Regressions: Quadratic Speedups without Data-Dependent Parameters
Keywords: Linear regression, quantum algorithms
TL;DR: We design a quantum algorithm solving the linear regression problem that does not require the data-dependent parameters.
Abstract: Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b$, the goal is to find $x'$ such that $\|| Ax' - b \||\_2^2 \leq (1+\epsilon) \min\_{x} \|| A x - b \||\_2^2$. The best classical algorithm takes $O(nd) + \mathrm{poly}(d/\epsilon)$ time [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]. On the other hand, quantum linear regression algorithms can achieve exponential quantum speedups, as shown in [Wang \emph{Phys. Rev. A 96}, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{\'e}n and Jeffery ICALP 2019]. However, the running times of these algorithms depend on some quantum linear algebra-related parameters, such as $\kappa(A)$, the condition number of $A$. In this work, we develop a quantum algorithm that runs in $\widetilde{O}(\epsilon^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/\epsilon)$ time and outputs a classical solution. It provides a quadratic quantum speedup in $n$ over the classical lower bound without any dependence on data-dependent parameters. In addition, we also show our result can be generalized to multiple regression and ridge linear regression.
Primary Area: optimization
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Submission Number: 7724
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