A simple analysis of a quantum-inspired algorithm for solving low-rank linear systems

We describe and analyze a simple algorithm for sampling from the solution $\mathbf{x}^* := \mathbf{A}^+\mathbf{b}$ to a linear system $\mathbf{A}\mathbf{x} = \mathbf{b}$. We assume access to a sampler which allows us to draw indices proportional to the squared row/column-norms of $\mathbf{A}$. Our algorithm produces a compressed representation of some vector $\mathbf{x}$ for which $\|\mathbf{x}^* - \mathbf{x}\| < \varepsilon \|\mathbf{x}^* \|$ in $\widetilde{O}(κ_{\mathsf{F}}^4 κ^2 / \varepsilon^2)$ time, where $κ_{\mathsf{F}} := \|\mathbf{A}\|_{\mathsf{F}}\|\mathbf{A}^{+}\|$ and $κ:= \|\mathbf{A}\|\|\mathbf{A}^{+}\|$. The representation of $\mathbf{x}$ allows us to query entries of $\mathbf{x}$ in $\widetilde{O}(κ_{\mathsf{F}}^2)$ time and sample proportional to the square entries of $\mathbf{x}$ in $\widetilde{O}(κ_{\mathsf{F}}^4 κ^6)$ time, assuming access to a sampler which allows us to draw indices proportional to the squared entries of any given row of $\mathbf{A}$. Our analysis, which is elementary, non-asymptotic, and fully self-contained, simplifies and clarifies several past analyses from literature including [Gilyén, Song, and Tang; 2022, 2023] and [Shao and Montanaro; 2022].