We give a classical analogue to Kerenidis and Prakash's quantum
recommendation system, previously believed to be one of the strongest
candidates for provably exponential speedups in quantum machine learning. Our
main result is an algorithm that, given an $m \times n$ matrix in a data
structure supporting certain $\ell^2$-norm sampling operations, outputs an
$\ell^2$-norm sample from a rank-$k$ approximation of that matrix in time
$O(\text{poly}(k)\log(mn))$, only polynomially slower than the quantum
algorithm. As a consequence, Kerenidis and Prakash's algorithm does not in fact
give an exponential speedup over classical algorithms. Further, under strong
input assumptions, the classical recommendation system resulting from our
algorithm produces recommendations exponentially faster than previous classical
systems, which run in time linear in $m$ and $n$.
The main insight of this work is the use of simple routines to manipulate
$\ell^2$-norm sampling distributions, which play the role of quantum
superpositions in the classical setting. This correspondence indicates a
potentially fruitful framework for formally comparing quantum machine learning
algorithms to classical machine learning algorithms.

通过使用简单例程来操纵 $\ell^2$ -norm 抽样分布，并提供一个给定数据结构的算法来生成从矩阵秩 - $k$ 近似值的 $\ell^2$ -norm 样本，本研究为 Kerenidis 和 Prakash 的量子推荐系统提供了古典类比，进一步证明此算法并未超越经典算法；此外，尽管假定输入数据重程度较大，但我们的经典推荐系统与先前线性 m 和 n 时间运行的经典系统相比能够在指数上更快地产生推荐。