  We present an algorithmic framework for quantum-inspired classical algorithms
on close-to-low-rank matrices, generalizing the series of results started by
Tang's breakthrough quantum-inspired algorithm for recommendation systems
[STOC'19]. Motivated by quantum linear algebra algorithms and the quantum
singular value transformation (SVT) framework of Gily\'en, Su, Low, and Wiebe
[STOC'19], we develop classical algorithms for SVT that run in time independent
of input dimension, under suitable quantum-inspired sampling assumptions. Our
results give compelling evidence that in the corresponding QRAM data structure
input model, quantum SVT does not yield exponential quantum speedups. Since the
quantum SVT framework generalizes essentially all known techniques for quantum
linear algebra, our results, combined with sampling lemmas from previous work,
suffice to generalize all recent results about dequantizing quantum machine
learning algorithms. In particular, our classical SVT framework recovers and
often improves the dequantization results on recommendation systems, principal
component analysis, supervised clustering, support vector machines, low-rank
regression, and semidefinite program solving. We also give additional
dequantization results on low-rank Hamiltonian simulation and discriminant
analysis. Our improvements come from identifying the key feature of the
quantum-inspired input model that is at the core of all prior quantum-inspired
results: $\ell^2$-norm sampling can approximate matrix products in time
independent of their dimension. We reduce all our main results to this fact,
making our exposition concise, self-contained, and intuitive.
