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 decomposition, our method achieves lower asymptotic complexity per solve than recent state-of-the-art methods. Experiments on billion-scale sparse tensors and synthetic data validate our theoretical claims, with our algorithm achieving higher accuracy than competing methods as the decomposition rank grows.

翻译：我们提出一种数据结构，用于根据多个矩阵的Khatri-Rao乘积的杠杆分数精确分布对其行进行随机采样。所提出的采样器以Khatri-Rao乘积高度对数时间、列数平方时间抽取每一行，同时最大持久空间开销不超过输入矩阵的规模。因此，即使构成Khatri-Rao乘积的矩阵每行各有数千万行，该采样器也能高效地进行采样。当该方法用于对Candecomp/PARAFAC分解中的线性最小二乘问题进行草图化处理时，每次求解的渐近复杂度低于当前最新方法。在十亿级稀疏张量和合成数据上的实验结果验证了我们的理论主张：随着分解秩的增加，我们的算法相比竞争方法实现了更高的精度。