An improved quantum-inspired algorithm for linear regression

We give a classical algorithm for linear regression analogous to the quantum matrix inversion [Harrow, Hassidim, and Lloyd, Physical Review Letters'09, arXiv:0811.3171] low-rank matrices [Wossnig, Zhao, Prakash, Letters'18, arXiv:1704.06174], when input $A$ is stored in data structure applicable QRAM-based state preparation. Namely, suppose we are given an $A \in \mathbb{C}^{m\times n}$ with minimum non-zero singular value $\sigma$ which supports certain efficient $\ell_2$-norm importance sampling queries, along $b \mathbb{C}^m$. Then, some $x \mathbb{C}^n$ satisfying $\|x - A^+b\| \leq \varepsilon\|A^+b\|$, can output measurement of $|x\rangle$ computational basis entry $x$ algorithms that run $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^6}{\sigma^{12}\varepsilon^4}\big)$ $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^2}{\sigma^8\varepsilon^4}\big)$ time, respectively. This improves on previous "quantum-inspired" this line research by at least factor $\frac{\|A\|^{16}}{\sigma^{16}\varepsilon^2}$ [Chia, Gily\'en, Li, Lin, Tang, Wang, STOC'20, arXiv:1910.06151]. As consequence, show computers achieve most factor-of-12 speedup QRAM setting related settings. Our work applies techniques from sketching optimization quantum-inspired literature. Unlike earlier works, promising avenue could lead feasible implementations settings, comparison against future computers.