A high order solver for signature kernels
Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differe...
| Main Authors: | , |
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| Format: | Internet publication |
| Language: | English |
| Published: |
2024
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| _version_ | 1811139631218950144 |
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| author | Lemercier, M Lyons, T |
| author_facet | Lemercier, M Lyons, T |
| author_sort | Lemercier, M |
| collection | OXFORD |
| description | Signature kernels are at the core of several machine learning algorithms for analysing
multivariate time series. The kernel of two bounded variation paths (such as piecewise
linear interpolations of time series data) is typically computed by solving a Goursat
problem for a hyperbolic partial differential equation (PDE) in two independent time
variables. However, this approach becomes considerably less practical for highly oscillatory
input paths, as they have to be resolved at a fine enough scale to accurately recover
their signature kernel, resulting in significant time and memory complexities. To mitigate
this issue, we first show that the signature kernel of a broader class of paths, known
as smooth rough paths, also satisfies a PDE, albeit in the form of a system of coupled
equations.We then use this result to introduce new algorithms for the numerical approximation
of signature kernels. As bounded variation paths (and more generally geometric
p-rough paths) can be approximated by piecewise smooth rough paths, one can replace
the PDE with rapidly varying coefficients in the original Goursat problem by an explicit
system of coupled equations with piecewise constant coefficients derived from the first
few iterated integrals of the original input paths. While this approach requires solving
more equations, they do not require looking back at the complex and fine structure of
the initial paths, which significantly reduces the computational complexity associated
with the analysis of highly oscillatory time series. |
| first_indexed | 2024-09-25T04:09:09Z |
| format | Internet publication |
| id | oxford-uuid:924f1e49-7953-4518-8f2b-6f5ed8a6a230 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:09:09Z |
| publishDate | 2024 |
| record_format | dspace |
| spelling | oxford-uuid:924f1e49-7953-4518-8f2b-6f5ed8a6a2302024-06-12T13:22:07ZA high order solver for signature kernelsInternet publicationhttp://purl.org/coar/resource_type/c_7ad9uuid:924f1e49-7953-4518-8f2b-6f5ed8a6a230EnglishSymplectic Elements2024Lemercier, MLyons, TSignature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as smooth rough paths, also satisfies a PDE, albeit in the form of a system of coupled equations.We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric p-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series. |
| spellingShingle | Lemercier, M Lyons, T A high order solver for signature kernels |
| title | A high order solver for signature kernels |
| title_full | A high order solver for signature kernels |
| title_fullStr | A high order solver for signature kernels |
| title_full_unstemmed | A high order solver for signature kernels |
| title_short | A high order solver for signature kernels |
| title_sort | high order solver for signature kernels |
| work_keys_str_mv | AT lemercierm ahighordersolverforsignaturekernels AT lyonst ahighordersolverforsignaturekernels AT lemercierm highordersolverforsignaturekernels AT lyonst highordersolverforsignaturekernels |