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

2025-08-18

AI Summary

This paper addresses the least-squares solution $x^*$ of an ill-conditioned low-rank linear system $Ax = b$. We propose a quantum-inspired classical sampling algorithm that efficiently produces a compressed representation of $x^*$, supporting both entry-wise queries and probability sampling. The method employs $ell_2$-norm squared sampling over rows and columns of $A$, and its analysis is non-asymptotic, self-contained, and elementary—parameterized by the Frobenius condition number $kappa_F$ and spectral condition number $kappa$. Theoretically, it outputs an $varepsilon$-accurate approximation $x$ satisfying $|x - x^*| < varepsilon |x^*|$ in $ ilde{O}(kappa_F^4 kappa^2 / varepsilon^2)$ time; supports entry queries in $ ilde{O}(kappa_F^2)$ time; and enables squared-weight sampling in $ ilde{O}(kappa_F^4 kappa^6)$ time. Our main contribution is a simplified, accessible, rigorous, and practically grounded classical sampling framework inspired by quantum algorithms—bridging theoretical guarantees with implementable efficiency.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

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].

Problem

• Sampling from low-rank linear system solutions efficiently
• Producing compressed representations with bounded approximation error
• Enabling fast querying and sampling from solution vectors

Innovation

• Quantum-inspired algorithm for low-rank linear systems
• Sampler access for squared row/column-norm indices
• Compressed representation enabling efficient querying and sampling