We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09] for low-rank matrices [Wossnig et al., Physical Review Letters'18], when the input matrix $A$ is stored in a data structure applicable for QRAM-based state preparation. Namely, given an $A \in \mathbb{C}^{m\times n}$ with minimum singular value $\sigma$ and which supports certain efficient $\ell_2$-norm importance sampling queries, along with a $b \in \mathbb{C}^m$, we can output a description of an $x \in \mathbb{C}^n$ such that $\|x - A^+b\| \leq \varepsilon\|A^+b\|$ in $\tilde{\mathcal{O}}\Big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^2}{\sigma^8\varepsilon^4}\Big)$ time, improving on previous "quantum-inspired" algorithms in this line of research by a factor of $\frac{\|A\|^{14}}{\sigma^{14}\varepsilon^2}$ [Chia et al., STOC'20]. The algorithm is stochastic gradient descent, and the analysis bears similarities to those of optimization algorithms for regression in the usual setting [Gupta and Sidford, NeurIPS'18]. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired setting, for comparison against future quantum computers.

中文翻译：

---

一种用于线性回归的改进的量子启发算法

我们给出了线性回归的经典算法，类似于用于低秩矩阵的量子矩阵求逆算法 [Harrow, Hassidim, and Lloyd, Physical Review Letters'09] [Wossnig et al., Physical Review Letters'18]，当输入矩阵 $A$ 存储在适用于基于 QRAM 的状态准备的数据结构中。即，给定一个 $A \in \mathbb{C}^{m\times n}$ 具有最小奇异值 $\sigma$ 并且支持某些有效的 $\ell_2$-norm 重要性采样查询，以及 $b \在 \mathbb{C}^m$ 中，我们可以输出 $x \in \mathbb{C}^n$ 的描述，使得 $\|x - A^+b\| \leq \varepsilon\|A^+b\|$ in $\tilde{\mathcal{O}}\Big(\frac{\|A\|_{\mathrm{F}}^6\|A\| ^2}{\sigma^8\varepsilon^4}\Big)$ 时间，改进了之前的“量子启发” 这一系列研究中的算法的因子为 $\frac{\|A\|^{14}}{\sigma^{14}\varepsilon^2}$ [Chia et al., STOC'20]。该算法是随机梯度下降算法，其分析与通常设置中的回归优化算法相似 [Gupta 和 Sidford，NeurIPS'18]。与早期的工作不同，这是一条很有前途的途径，可以在受量子启发的环境中实现经典回归的可行实现，以便与未来的量子计算机进行比较。