mlsanityData to 14 Oct 2026

Log-PDE Methods for Rough Signature Kernels

arXiv 2404.02926 ↗First posted 1 Apr 2024cs.LG math.APcited by 0

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Abstract (arXiv)

Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However, existing PDE solvers only use increments as input data, leading to first order approximation errors. These approaches become computationally intractable 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. In this paper, we extend the analysis to rough paths, and show, leveraging the framework of smooth rough paths, that the resulting rough signature kernels can be approximated by a novel system of PDEs whose coefficients involve higher order iterated integrals of the input rough paths. We show that this system of PDEs admits a unique solution and establish quantitative error bounds yielding a higher order approximation to rough signature kernels.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 25 of its 58 references that are in the map. First posted on arXiv 1 Apr 2024.

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25 of 58 references.

References · 25

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  2. 2017Deep Neural Networks as Gaussian Processescited by 622
  3. 2017MMD GAN: Towards Deeper Understanding of Moment Matching Networkcited by 254
  4. 2020Neural Rough Differential Equations for Long Time Seriescited by 143
  5. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  6. 2016Kernels for sequentially ordered datacited by 84
  7. 2019Deep Signature Transformscited by 75
  8. 2024Theoretical Foundations of Deep Selective State-Space Modelscited by 74
  9. 2020The Signature Kernel is the solution of a Goursat PDEcited by 58
  10. 2021Robust Generalised Bayesian Inference for Intractable Likelihoodscited by 53
  11. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  12. 2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 47
13 more
  1. 2020Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPUcited by 46
  2. 2018The iisignature library: efficient calculation of iterated-integral signatures and log signaturescited by 40
  3. 2023Non-adversarial training of Neural SDEs with signature kernel scorescited by 37
  4. 2020Distribution Regression for Sequential Datacited by 31
  5. 2020A Kernel Two-Sample Test for Functional Datacited by 28
  6. 2021Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processescited by 27
  7. 2023Neural signature kernels as infinite-width-depth-limits of controlled ResNetscited by 25
  8. 2021Framing RNN as a kernel method: A neural ODE approachcited by 23
  9. 2021SigGPDE: Scaling Sparse Gaussian Processes on Sequential Datacited by 22
  10. 2024Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processescited by 18
  11. 2021SK-Tree: a systematic malware detection algorithm on streaming trees via the signature kernelcited by 16
  12. 2023A Neural RDE approach for continuous-time non-Markovian stochastic control problemscited by 14
  13. 2023New directions in the applications of rough path theorycited by 12

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