  Tensor algebras give rise to one of the most powerful measures of similarity
for sequences of arbitrary length called the signature kernel accompanied with
attractive theoretical guarantees from stochastic analysis. Previous algorithms
to compute the signature kernel scale quadratically in terms of the length and
the number of the sequences. To mitigate this severe computational bottleneck,
we develop a random Fourier feature-based acceleration of the signature kernel
acting on the inherently non-Euclidean domain of sequences. We show uniform
approximation guarantees for the proposed unbiased estimator of the signature
kernel, while keeping its computation linear in the sequence length and number.
In addition, combined with recent advances on tensor projections, we derive two
even more scalable time series features with favourable concentration
properties and computational complexity both in time and memory. Our empirical
results show that the reduction in computational cost comes at a negligible
price in terms of accuracy on moderate-sized datasets, and it enables one to
scale to large datasets up to a million time series.
