Rough paths, Signatures and the modelling of functions on streams

Abstract

Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It draws on the analysis of LC Young and the geometric algebra of KT Chen. The concepts and the uniform estimates, have widespread application and have simplified proofs of basic questions from the large deviation theory and extended Ito's theory of SDEs; the recent applications contribute to (Graham) automated recognition of Chinese handwriting and (Hairer) formulation of appropriate SPDEs to model randomly evolving interfaces. At the heart of the mathematics is the challenge of describing a smooth but potentially highly oscillatory and vector valued path $x_t$ parsimoniously so as to effectively predict the response of a nonlinear system such as $d{y}_t=f(y_t)d{x}_t$, $y_0=a$. The Signature is a homomorphism from the monoid of paths into the grouplike elements of a closed tensor algebra. It provides a graduated summary of the path $x$. Hambly and Lyons have shown that this non-commutative transform is faithful for paths of bounded variation up to appropriate null modifications. Among paths of bounded variation with given Signature there is always a unique shortest representative. These graduated summaries or features of a path are at the heart of the definition of a rough path; locally they remove the need to look at the fine structure of the path. Taylor's theorem explains how any smooth function can, locally, be expressed as a linear combination of certain special functions (monomials based at that point). Coordinate iterated integrals form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough path and a natural linear "basis" for functions on streams that can be used for machine learning.

arXiv commentTo appear in the Proceedings of the International Congress of Mathematicians 2014, Korea

References in corpus (8)

• Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
• Learning from the past, predicting the statistics for the future, learning an evolving system
• Extracting information from the signature of a financial data stream
• Discretely sampled signals and the rough Hoff process
• Evolving Communities with Individual Preferences
• Dimension-free Euler estimates of rough differential equations
• The partial sum process of orthogonal expansion as geometric rough process with Fourier series as an example---an improvement of Menshov-Rademacher theorem
• Uniqueness of signature for simple curves

Cited by in corpus (30)

• Neural Controlled Differential Equations for Irregular Time Series
• Characteristic functions of measures on geometric rough paths
• Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU
• Deep Signature Transforms
• Discretely sampled signals and the rough Hoff process
• Using path signatures to predict a diagnosis of Alzheimer's disease
• Deriving information from missing data: implications for mood prediction
• Neural Rough Differential Equations for Long Time Series
• Fully Convolutional Recurrent Network for Handwritten Chinese Text Recognition
• Detecting early signs of depressive and manic episodes in patients with bipolar disorder using the signature-based model
• Pathwise approximation of SDEs by coupling piecewise abelian rough paths
• Framing RNN as a kernel method: A neural ODE approach
• Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes
• Understanding Recurrent Neural Networks Using Nonequilibrium Response Theory
• Distribution Regression for Sequential Data
• Pathwise Itô Calculus for Rough Paths and Rough PDEs with Path Dependent Coefficients
• Seq2Tens: An Efficient Representation of Sequences by Low-Rank Tensor Projections
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
• Sketching the order of events
• SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data
• Signature features with the visibility transformation
• Learning to Detect Bipolar Disorder and Borderline Personality Disorder with Language and Speech in Non-Clinical Interviews
• Approximate Bayesian Computation with Path Signatures
• Learning Spatial-Semantic Context with Fully Convolutional Recurrent Network for Online Handwritten Chinese Text Recognition
• Path Imputation Strategies for Signature Models of Irregular Time Series
• Path Signatures on Lie Groups
• Robust Hedging GANs
• Path Signature Area-Based Causal Discovery in Coupled Time Series
• DeepWriterID: An End-to-end Online Text-independent Writer Identification System
• Deep Learning for Estimating Synaptic Health of Primary Neuronal Cell Culture