Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension

TL;DR

This paper presents a classical algorithm inspired by quantum techniques for low-rank matrix inversion, achieving logarithmic dependence on the dimension and enabling efficient sampling-based solutions.

Contribution

It introduces a classical analogue of the quantum matrix inversion algorithm for low-rank matrices using sampling and SVD approximation, facilitating dequantization of quantum algorithms.

Findings

Achieves logarithmic dependence on dimension for low-rank matrix inversion

Provides a sampling-based method to implement pseudoinverses and apply functions to singular values

Enables classical dequantization of certain quantum algorithms

Abstract

We construct an efficient classical analogue of the quantum matrix inversion algorithm (HHL) for low-rank matrices. Inspired by recent work of Tang, assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix and sample from the solution to the problem $Ax=b$ using fast sampling techniques. We implement the pseudo-inverse by finding an approximate singular value decomposition of $A$ via subsampling, then inverting the singular values. In principle, the approach can also be used to apply any desired "smooth" function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem, our result suggests that more low-rank quantum algorithms can be effectively "dequantised" into classical length-square sampling algorithms.