2018

Signature moments to characterize laws of stochastic processes

Chevyrev, Ilya, Oberhauser, Harald

Understand

The sequence of moments of a vector-valued random variable can characterize its law.

We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments.
This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic processes and study the topology it induces on the space of laws of stochastic processes.
This metric can be kernelized using the signature kernel which allows to efficiently compute it.

Built on

A generalization of the strict topology
Robin Giles · 1971
Earlier work this paper cites.
Algorithms for minimization without derivatives
Richard P. Brent · 1973
Earlier work this paper cites.
Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests
Jerome H Friedman and Lawrence C Rafsky · 1979
Earlier work this paper cites.
Robust statistics: the approach based on influence functions
Frank R. Hampel, Elvezio M. Ronchetti, Peter J. Rousseeuw, and Werner A. Stahel · 1986
Earlier work this paper cites.
Probability metrics and the stability of stochastic models , volume 269
Svetlozar Todorov Rachev · 1991
Earlier work this paper cites.
Free Lie algebras
Christophe Reutenauer · 1993
Earlier work this paper cites.
Using dynamic time warping to find patterns in time series
Donald J Berndt and James Clifford · 1994
Earlier work this paper cites.
Integral probability metrics and their generating classes of functions
Alfred Müller · 1997
Earlier work this paper cites.
Differential equations driven by rough signals
Terry Lyons · 1998
Earlier work this paper cites.
SciPy: Open source scientific tools for Python, 2001–
Eric Jones, Travis Oliphant, and Pearu Peterson · 2001
Earlier work this paper cites.
Extending the Wong-Zakai theorem to reversible Markov processes
Richard Bass, Ben Hambly, and Terry Lyons · 2002
Earlier work this paper cites.
Stochastic analysis, rough path analysis and fractional Brownian motions
Laure Coutin and Zhongmin Qian · 2002
Earlier work this paper cites.
On the mathematical foundations of learning
Felipe Cucker and Steve Smale · 2002
Earlier work this paper cites.
Algebra
Serge Lang · 2002
Earlier work this paper cites.
System control and rough paths
Terry Lyons and Zhongmin Qian · 2002
Earlier work this paper cites.
Learning with kernels: support vector machines, regularization, optimization, and beyond
Bernhard Schölkopf and Alexander J Smola · 2002
Earlier work this paper cites.
Problems in stochastic analysis: connections between rough paths and non-commutative harmonic analysis
Thomas Fawcett · 2003
Earlier work this paper cites.
A Kullback-Leibler Divergence Based Kernel for SVM Classification in Multimedia Applications
Pedro J Moreno, Purdy Ho, and Nuno Vasconcelos · 2003
Earlier work this paper cites.

Similar

Probabilistic kernels for the classification of auto-regressive visual processes
Antoni B Chan and Nuno Vasconcelos · 2005
Cited alongside, same era.
Testing statistical hypotheses
Erich L Lehmann and Joseph P Romano · 2006
Cited alongside, same era.
Differential equations driven by rough paths , volume 1908 of Lecture Notes in Mathematics
Terry Lyons, Michael Caruana, and Thierry Lévy · 2007
Cited alongside, same era.
On uniformly subelliptic operators and stochastic area
Peter Friz and Nicolas Victoir · 2008
Cited alongside, same era.
Kernel choice and classifiability for rkhs embeddings of probability distributions
Kenji Fukumizu, Arthur Gretton, Gert R Lanckriet, Bernhard Schölkopf, and Bharath K Sriperumbudur · 2009
Cited alongside, same era.
A fast, consistent kernel two-sample test
Arthur Gretton, Kenji Fukumizu, Zaid Harchaoui, and Bharath K Sriperumbudur · 2009
Cited alongside, same era.
Robust statistics
Peter J. Huber and Elvezio M. Ronchetti · 2009
Cited alongside, same era.
Multidimensional stochastic processes as rough paths , volume 120 of Cambridge Studies in Advanced Mathematics
Peter K. Friz and Nicolas B. Victoir · 2010
Cited alongside, same era.
Hilbert space embeddings and metrics on probability measures
Bharath K Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Bernhard Schölkopf, and Gert RG Lanckriet · 2010
Cited alongside, same era.
Reproducing kernel Hilbert spaces in probability and statistics
Alain Berlinet and Christine Thomas-Agnan · 2011
Cited alongside, same era.
Autoregressive kernels for time series
Original
Marco Cuturi and Arnaud Doucet · 2011
Cited alongside, same era.
Parameter estimation for rough differential equations
Anastasia Papavasiliou and Christophe Ladroue · 2011
Cited alongside, same era.

Then

On the integration of weakly geometric rough paths
Thomas Cass, Bruce K. Driver, Nengli Lim, and Christian Litterer · 2016
Later among the works it cites.
A Primer on the Signature Method in Machine Learning
Ilya Chevyrev and Andrey Kormilitzin · 2016
Later among the works it cites.
Characteristic Functions of Measures on Geometric Rough Paths
Ilya Chevyrev and Terry Lyons · 2016
Later among the works it cites.
The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory
Peter K. Friz, Benjamin Gess, Archil Gulisashvili, and Sebastian Riedel · 2016
Later among the works it cites.
On the optimal estimation of probability measures in weak and strong topologies
Bharath Sriperumbudur · 2016
Later among the works it cites.
Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy
Original
Dougal J. Sutherland, Hsiao-Yu Tung, Heiko Strathmann, Soumyajit De, Aaditya Ramdas, Alex Smola, and Arthur Gretton · 2016
Later among the works it cites.
General rough integration, Lévy rough paths and a Lévy–Kintchine type formula
Peter Friz and Atul Shekhar · 2017
Later among the works it cites.
Kernel mean embedding of distributions: A review and beyond
Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, and Bernhard Schölkopf · 2017
Later among the works it cites.
The UEA multivariate time series classification archive, 2018
Anthony Bagnall, Hoang Anh Dau, Jason Lines, Michael Flynn, James Large, Aaron Bostrom, Paul Southam, and Eamonn Keogh · 2018
Closest in time.
Random walks and Lévy processes as rough paths
Ilya Chevyrev · 2018
Closest in time.
A support and density theorem for Markovian rough paths
Ilya Chevyrev and Marcel Ogrodnik · 2018
Closest in time.
Kernel distribution embeddings: Universal kernels, characteristic kernels and kernel metrics on distributions
Carl-Johann Simon-Gabriel and Bernhard Schölkopf · 2018
Closest in time.
Canonical RDEs and general semimartingales as rough paths
Ilya Chevyrev and Peter K. Friz · 2019
Closest in time.
Kernels for sequentially ordered data
Franz J. Kiraly and Harald Oberhauser · 2019
Closest in time.
Persistence Paths and Signature Features in Topological Data Analysis
I. Chevyrev, V. Nanda, and H. Oberhauser · 2020
Closest in time.
Metrizing Weak Convergence with Maximum Mean Discrepancies
Original
Carl-Johann Simon-Gabriel, Alessandro Barp, and Lester Mackey · 2020
Closest in time.
Bayesian Learning from Sequential Data using Gaussian Processes with Signature Covariances
Csaba Toth and Harald Oberhauser · 2020
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…