Sparse Signature Coefficient Recovery via Kernels

TL;DR

This paper introduces a novel method leveraging signature kernels and PDE-based techniques to efficiently identify and recover sparse collections of signature coefficients in high levels of the signature transform, enhancing computational efficiency.

Contribution

It presents a new approach for sparse signature coefficient recovery using kernel filters and PDE methods, addressing computational challenges in high-level signature analysis.

Findings

Effective isolation of specific signature coefficients demonstrated empirically

Kernel-based filters can be expressed as linear combinations of signature transforms

Application shown in constructing sparse Euler schemes for controlled differential equations

Abstract

Central to rough path theory is the signature transform of a path, an infinite series of tensors given by the iterated integrals of the underlying path. The signature poses an effective way to capture sequentially ordered information, thanks both to its rich analytic and algebraic properties as well as its universality when used as a basis to approximate functions on path space. Whilst a truncated version of the signature can be efficiently computed using Chen's identity, there is a lack of efficient methods for computing a sparse collection of iterated integrals contained in high levels of the signature. We address this problem by leveraging signature kernels, defined as the inner product of two signatures, and computable efficiently by means of PDE-based methods. By forming a filter in signature space with which to take kernels, one can effectively isolate specific groups of signature coefficients and, in particular, a singular coefficient at any depth of the transform. We show that such a filter can be expressed as a linear combination of suitable signature transforms and demonstrate empirically the effectiveness of our approach. To conclude, we give an example use case for sparse collections of signature coefficients based on the construction of N-step Euler schemes for sparse CDEs.