  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.
