Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes

Salvi, Cristopher; Lemercier, Maud; Liu, Chong; Hovarth, Blanka; Damoulas, Theodoros; Lyons, Terry

doi:10.48550/arxiv.2109.03582

Cited by 2 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One notable example of a jointly controlled path satisfying Condition 3.6 is the signature kernel, a machine learning tool which has recently seen use in kernel methods in works such as [4,13,14,[16][17][18]. An important result for the computation of the signature kernel is that it satisfies a second order Goursat PDE in the case where paths are differentiable [17].…”

Section: Signature Kernels As Jointly Controlled Pathsmentioning

confidence: 99%

See 1 more Smart Citation

A Fubini type theorem for rough integration

Cass

Pei

2023

Rev. Mat. Iberoam.

View full text Add to dashboard Cite

Jointly controlled paths as used in Gerasimovičs and Hairer (2019), are a class of two-parameter paths Y controlled by a p-rough path X for 2 Ä p < 3 in each time variable, and serve as a class of paths twice integrable with respect to X. We extend the notion of jointly controlled paths to two-parameter paths Y controlled by p-rough and Q p-rough paths X and Q X (on finite dimensional spaces) for arbitrary p and Q p, and develop the corresponding integration theory for this class of paths. In particular, we show that for paths Y jointly controlled by X and Q X , they are integrable with respect to X and Q X, and moreover we prove a rough Fubini type theorem for the double rough integrals of Y via the construction of a third integral analogous to the integral against the product measure in the classical Fubini theorem. Additionally, we also prove a stability result for the double integrals of jointly controlled paths, and show that signature kernels, which have seen increasing use in data science applications, are jointly controlled paths.

show abstract

Section: Signature Kernels As Jointly Controlled Pathsmentioning

confidence: 99%

“…The second is the signature kernel, which has seen data science applications in works such as [4,13,14,[16][17][18]. This also gives us an alternative account of giving meaning to the rough integral equation " K .s;t /;.u;v/ .X;…”

Section: Introductionmentioning

confidence: 99%

A Fubini type theorem for rough integration

Cass

Pei

2023

Rev. Mat. Iberoam.

View full text Add to dashboard Cite

show abstract

“…These results have been influential in stochastic analysis [14,15] and only more recently have been started to be explored in a statistical and machine learning context. We mention pars-pro-toto [16,17,18] for inference about laws of stochastic processes; [10,19,20] for kernel learning; [21,22,23] for Bayesian approaches; [24,25,26] for generative modelling; [27,28,29] for applications in topology; [30,31,32] for algebraic perspectives.…”

Section: Related Workmentioning

confidence: 99%

“…However, for some applications the filtration of a stochastic process matters and one could ask to extend the scoring rule framework to this. A kernel that captures the filtration was introduced in [50] and a kernel algorithm and new applications given in [19]. To get a non-kernel scoring one could try to replace Φ in Definition (4.1) by the higher-rank signature from [50].…”

Section: Scoring Rules For Tracks and Pathsmentioning

confidence: 99%

Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Series

Bonnier¹,

Oberhauser²

2021

Preprint

View full text Add to dashboard Cite

Many forecasts consist not of point predictions but concern the evolution of quantities. For example, a central bank might predict the interest rates during the next quarter, an epidemiologist might predict trajectories of infection rates, a clinician might predict the behaviour of medical markers over the next day, etc. The situation is further complicated since these forecasts sometimes only concern the approximate "shape of the future evolution" or "order of events". Formally, such forecasts can be seen as probability measures on spaces of equivalence classes of paths modulo time-parametrization. We leverage the statistical framework of proper scoring rules with classical mathematical results to derive a principled approach to decision making with such forecasts. In particular, we introduce notions of gradients, entropy, and divergence that are tailor-made to respect the underlying non-Euclidean structure.

show abstract

Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes

Cited by 2 publications

References 0 publications

A Fubini type theorem for rough integration

A Fubini type theorem for rough integration

Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Series

Contact Info

Product

Resources

About