Random Fourier Signature Features

TL;DR

This paper introduces a random Fourier feature-based method to efficiently approximate the signature kernel for sequences, significantly reducing computational costs while maintaining accuracy, enabling large-scale time series analysis.

Contribution

It develops a novel unbiased estimator for the signature kernel using random Fourier features, achieving linear complexity in sequence length and dataset size.

Findings

Approximation guarantees for the estimator.

Scalability to datasets with up to a million time series.

Negligible accuracy loss with reduced computational cost.

Abstract

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.