  Approximating kernel functions with random features (RFs)has been a
successful application of random projections for nonparametric estimation.
However, performing random projections presents computational challenges for
large-scale problems. Recently, a new optical hardware called Optical
Processing Unit (OPU) has been developed for fast and energy-efficient
computation of large-scale RFs in the analog domain. More specifically, the OPU
performs the multiplication of input vectors by a large random matrix with
complex-valued i.i.d. Gaussian entries, followed by the application of an
element-wise squared absolute value operation - this last nonlinearity being
intrinsic to the sensing process. In this paper, we show that this operation
results in a dot-product kernel that has connections to the polynomial kernel,
and we extend this computation to arbitrary powers of the feature map.
Experiments demonstrate that the OPU kernel and its RF approximation achieve
competitive performance in applications using kernel ridge regression and
transfer learning for image classification. Crucially, thanks to the use of the
OPU, these results are obtained with time and energy savings.
