An improved quantum-inspired algorithm for linear regression

We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09, arXiv:0811.3171] for low-rank matrices [Wossnig, Zhao, and Prakash, Physical Review Letters'18, arXiv:1704.06174], when the input matrix A is stored in a data structure applicable for QRAM-based state preparation. Namely, suppose we are given an A ∈ mathbbCm× n with minimum non-zero singular value σ which supports certain efficient ℓ2-norm importance sampling queries, along with a b ∈ mathbbCm. Then, for some x ∈ mathbbCn satisfying |x - A^+b| ≤ varepsilon|A^+b|, we can output a measurement of |xrangle in the computational basis and output an entry of x with classical algorithms that run in tildeObig(frac|A|F6|A|6σ12varepsilon4big) and tildeObig(frac|A|F6|A|2σ8varepsilon4big) time, respectively. This improves on previous "quantum-inspired" algorithms in this line of research by at least a factor of frac|A|16σ16varepsilon2 [Chia, Gily'en, Li, Lin, Tang, and Wang, STOC'20, arXiv:1910.06151]. 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.