Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning

We present an algorithmic framework generalizing quantum-inspired polylogarithmic-time algorithms on low-rank matrices. Our work follows the line of research started by Tang's breakthrough classical algorithm for recommendation systems [STOC'19]. The main result of this work is an algorithm for singular value transformation on low-rank inputs in the quantum-inspired regime, where singular value transformation is a framework proposed by Gily\'en et al. [STOC'19] to study various quantum speedups. Since singular value transformation encompasses a vast range of matrix arithmetic, this result, combined with simple sampling lemmas from previous work, suffices to generalize all results dequantizing quantum machine learning algorithms to the authors' knowledge. Via simple black-box applications of our singular value transformation framework, we recover the dequantization results on recommendation systems, principal component analysis, supervised clustering, low-rank matrix inversion, low-rank semidefinite programming, and support vector machines. We also give additional dequantizations results on low-rank Hamiltonian simulation and discriminant analysis.