We present a data structure to randomly sample rows from the Khatri-Rao product of several matrices according to the exact distribution of its leverage scores. Our proposed sampler draws each row in time logarithmic in the height of the Khatri-Rao product and quadratic in its column count, with persistent space overhead at most the size of the input matrices. As a result, it tractably draws samples even when the matrices forming the Khatri-Rao product have tens of millions of rows each. When used to sketch the linear least squares problems arising in CANDECOMP / PARAFAC tensor decomposition, our method achieves lower asymptotic complexity per solve than recent state-of-the-art methods. Experiments on billion-scale sparse tensors validate our claims, with our algorithm achieving higher accuracy than competing methods as the decomposition rank grows.

翻译：我们提出了一种数据结构，可从多个矩阵的Khatri-Rao积中按照其杠杆分数的精确分布随机采样行。所提出的采样器以Khatri-Rao积高度的对数时间和列数的二次时间完成每行采样，持续空间开销不超过输入矩阵的规模。因此，即使构成Khatri-Rao积的矩阵各自拥有数千万行，该采样器仍能高效执行采样。当该方法用于CANDECOMP/PARAFAC张量分解中出现的线性最小二乘问题的草图化求解时，其单次求解的渐近复杂度低于当前最先进方法。在十亿级稀疏张量上的实验验证了我们的论断：随着分解秩的增长，该算法相比竞争方法实现了更高的精度。