Complex-to-Real Random Features for Polynomial Kernels

Polynomial kernels are among the most popular kernels in machine learning, since their feature maps model the interactions between the dimensions of the input data. However, these features correspond to tensor products of the input with itself, which makes their dimension grow exponentially with the polynomial degree. We address this issue by proposing Complexto-Real (CtR) sketches for tensor products that can be used as random feature approximations of polynomial kernels. These sketches leverage intermediate complex random projections, leading to better theoretical guarantees and potentially much lower variances than analogs using real projections. Our sketches are simple to construct and their final output is real-valued, which makes their downstream use straightforward. Finally, we show that they achieve state-of-the-art performance in terms of accuracy and speed.