An improved quantum-inspired algorithm for linear regression

Abstract

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, Zhao, and Prakash, Physical Review Letters'18], when the input matrix A is stored in a data structure applicable for QRAM-based state preparation. Namely, suppose we are given an A ∈ ℂm×n with minimum non-zero singular value σ which supports certain efficient ℓ2-norm importance sampling queries, along with a b ∈ ℂm. Then, for some x ∈ ℂn satisfying ||x - A+b|| ≤ ε||A+b||, we can output a measurement of |x〉 in the computational basis and output an entry of x with classical algorithms that run in (equation presented) and (equation presented) time, respectively. This improves on previous “quantum-inspired” algorithms in this line of research by at least a factor of (equation presented) [Chia, Gilyén, Li, Lin, Tang, and Wang, STOC'20]. As a consequence, we show that quantum computers can achieve at most a factor-of-12 speedup for linear regression in this QRAM data structure setting and related settings. Our work applies techniques from sketching algorithms and optimization to the quantum-inspired literature. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired settings, for comparison against future quantum computers.