mlsanityData to 7 Oct 2026

Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes

arXiv 2109.03582 ↗First posted 8 Sep 2021stat.ML cs.LGcited by 27

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Abstract (arXiv)

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causal-discovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). Builds on 19 of its 59 references that are in the map. First posted on arXiv 8 Sep 2021.

Builds on

19 of 59 references.

References · 19

  1. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 360
  2. 2020Learning Deep Kernels for Non-Parametric Two-Sample Testscited by 129
  3. 2018Deep Hedgingcited by 106
  4. 2016Generative Models and Model Criticism via Optimized Maximum Mean Discrepancycited by 99
  5. 2020A Measure-Theoretic Approach to Kernel Conditional Mean Embeddingscited by 86
  6. 2016Kernels for sequentially ordered datacited by 83
  7. 2020Causal Discovery in Physical Systems from Videoscited by 77
  8. 2019Deep Signature Transformscited by 75
  9. 2020The Signature Kernel is the solution of a Goursat PDEcited by 58
  10. 2018Signature moments to characterize laws of stochastic processescited by 54
  11. 2014Learning Theory for Distribution Regressioncited by 48
  12. 2020Distribution Regression for Sequential Datacited by 31
7 more
  1. 2021Optimal Stopping via Randomized Neural Networkscited by 26
  2. 2018Causal Inference via Kernel Deviance Measurescited by 22
  3. 2016Bayesian Learning of Kernel Embeddingscited by 15
  4. 2021Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysiscited by 12
  5. 2017Bayesian Approaches to Distribution Regressioncited by 9
  6. 2018Hyperparameter Learning for Conditional Kernel Mean Embeddings with Rademacher Complexity Boundscited by 4
  7. 2020Regularised Least-Squares Regression with Infinite-Dimensional Output Spacecited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 6; 2024: 11; 2025: 8; 2026: 1.

27 papers.

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  1. 2024Theoretical Foundations of Deep Selective State-Space Modelscited by 73
  2. 2023Non-adversarial training of Neural SDEs with signature kernel scorescited by 36
  3. 2023Neural signature kernels as infinite-width-depth-limits of controlled ResNetscited by 25
  4. 2024Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processescited by 18
  5. 2024Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patchingcited by 16
  6. 2025Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Modelscited by 16
  7. 2024Lecture notes on rough paths and applications to machine learningcited by 13
  8. 2023New directions in the applications of rough path theorycited by 12
  9. 2024Exact Gradients for Stochastic Spiking Neural Networks Driven by Rough Signalscited by 9
  10. 2023The Signature Kernelcited by 8
  11. 2024Dynamical Survival Analysis with Controlled Latent Statescited by 7
  12. 2024Sparse Signature Coefficient Recovery via Kernelscited by 5
15 more
  1. 2025pySigLib -- Fast Signature-Based Computations on CPU and GPUcited by 5
  2. 2022A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spacescited by 3
  3. 2024High Rank Path Development: an approach of learning the filtration of stochastic processescited by 3
  4. 2025ParallelFlow: Parallelizing Linear Transformers via Flow Discretizationcited by 3
  5. 2025Signature Maximum Mean Discrepancy Two-Sample Statistical Testscited by 3
  6. 2023Non-parametric online market regime detection and regime clustering for multidimensional and path-dependent data structurescited by 2
  7. 2024Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detectioncited by 1
  8. 2025Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groupscited by 1
  9. 2026Generating Financial Time Series by Matching Random Convolutional Featurescited by 1
  10. 2023Directed Chain Generative Adversarial Networkscited by 0
  11. 2024Log-PDE Methods for Rough Signature Kernelscited by 0
  12. 2024Scalable Signature-Based Distribution Regression via Reference Setscited by 0
  13. 2025Are Two Datasets Close Enough With Statistical Significance? A Kernel Distributional Closeness Testing Approachcited by 0
  14. 2025Novelty detection on path spacecited by 0
  15. 2025Random Controlled Differential Equationscited by 0