Randomized sketches of a tensor product of $p$ vectors follow a tradeoff between statistical efficiency and computational acceleration. Commonly used approaches avoid computing the high-dimensional tensor product explicitly, resulting in a suboptimal dependence of $\mathcal{O}(3^p)$ in the embedding dimension. We propose a simple Complex-to-Real (CtR) modification of well-known sketches that replaces real random projections by complex ones, incurring a lower $\mathcal{O}(2^p)$ factor in the embedding dimension. The output of our sketches is real-valued, which renders their downstream use straightforward. In particular, we apply our sketches to $p$-fold self-tensored inputs corresponding to the feature maps of the polynomial kernel. We show that our method achieves state-of-the-art performance in terms of accuracy and speed compared to other randomized approximations from the literature.

翻译：针对$p$个向量的张量积的随机草图在统计效率与计算加速之间存在权衡。常用方法避免显式计算高维张量积，导致嵌入维度出现$\mathcal{O}(3^p)$的次优依赖。我们提出对经典草图的简单复到实（CtR）修正，将实随机投影替换为复随机投影，从而将嵌入维度的因子降至$\mathcal{O}(2^p)$。我们草图的输出为实数值，便于后续直接应用。特别地，我们将所提草图应用于对应多项式核特征图的$p$重自张量输入。实验表明，与文献中其他随机近似方法相比，我们的方法在精度与速度上均达到最优性能。