Kernels for sequentially ordered data

We present a novel framework for kernel learning with sequential data of any kind, such as time series, sequences of graphs, or strings. Our approach is based on signature features which can be seen as an ordered variant of sample (cross-)moments; it allows to obtain a "sequentialized" version of any static kernel. The sequential kernels are efficiently computable for discrete sequences and are shown to approximate a continuous moment form in a sampling sense. A number of known kernels for sequences arise as "sequentializations" of suitable static kernels: string kernels may be obtained as a special case, and alignment kernels are closely related up to a modification that resolves their open non-definiteness issue. Our experiments indicate that our signature-based sequential kernel framework may be a promising approach to learning with sequential data, such as time series, that allows to avoid extensive manual pre-processing.

• Publication year

  2016

• Venue

  Journal of machine learning research

• Publication date

  2016-01-29

• Fields of study

  Mathematics, Computer Science

• Identifiers
  arXiv 1601.08169
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Statistics For Spatial Data

  2016cited by this paper
• The signature of a rough path: uniqueness

  2015cited by this paper
• Character-level Chinese Writer Identification using Path Signature Feature, DropStroke and Deep CNN

  2015cited by this paper
• A feature set for streams and an application to high-frequency financial tick data

  2014cited by this paper
• The Signature of a Rough Path: Uniqueness

  2014influential reference
• A feature set for streams and an application to high-frequency nancial tick data

  2014cited by this paper
• Extracting information from the signature of a financial data stream

  2013cited by this paper
• EMG based classification of basic hand movements based on time-frequency features

  2013influential reference
• Rotation invariants of two dimensional curves based on iterated integrals

  2013cited by this paper
• Learning from the past, predicting the statistics for the future, learning an evolving system

  2013cited by this paper
• Sparse arrays of signatures for online character recognition

  2013cited by this paper
• A theory of regularity structures

  2013cited by this paper
• Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups

  2012cited by this paper
• Fast Global Alignment Kernels

  2011influential reference
• Multidimensional Stochastic Processes as Rough Paths: Theory and Applications

  2010cited by this paper
• Computing and Visualizing Dynamic Time Warping Alignments in R: The dtw Package

  2009cited by this paper
• Parameter estimation for rough differential equations

  2008cited by this paper
• A generalization of Haussler's convolution kernel: mapping kernel

  2008influential reference
• Differential equations driven by rough paths

  2007influential reference
• A Kernel for Time Series Based on Global Alignments

  2006cited by this paper
• Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)

  2005cited by this paper
• Uniqueness for the signature of a path of bounded variation and the reduced path group

  2005influential reference
• Kernel Methods for Pattern Analysis

  2004cited by this paper
• Fast String Kernels using Inexact Matching for Protein Sequences

  2004influential reference
• Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond

  2003cited by this paper
• Text Classification using String Kernels

  2002influential reference
• Online handwriting recognition with support vector machines - a kernel approach

  2002influential reference
• On the mathematical foundations of learning

  2001cited by this paper
• Dynamic Time-Alignment Kernel in Support Vector Machine

  2001cited by this paper
• Text Classification using String Kernels

  2000influential reference
• Convolution kernels on discrete structures

  1999influential reference
• Di erential equations driven by rough signals

  1998cited by this paper
• In Advances in Neural Information Processing Systems

  1996cited by this paper
• Continuous martingales and Brownian motion

  1990cited by this paper
• A tutorial on hidden Markov models and selected applications in speech recognition

  1989cited by this paper
• An Overview of Sequence Comparison: Time Warps, String Edits, and Macromolecules

  1983cited by this paper
• Two-level DP-matching--A dynamic programming-based pattern matching algorithm for connected word recognition

  1979cited by this paper
• Principles of geostatistics

  1963cited by this paper

• Koopman Analysis of Sea Surface Temperature with a Signature Kernel

  2026cites this paper
• Signature-Kernel Based Evaluation Metrics for Robust Probabilistic and Tail-Event Forecasting

  2026cites this paper
• Nonparametric Teaching of Attention Learners

  2026cites this paper
• Kernel Integrated $R^2$: A Measure of Dependence

  2026cites this paper
• The Volterra signature

  2026cites this paper
• Manifold-regularised Signature Kernel Large-Margin ℓp-SVDD for Multidimensional Time Series Anomaly Detection

  2025cites this paper
• Structural changes and statistical causal relationships in agricultural commodities markets: the impact of public news sentiment and institutional announcements

  2025cites this paper
• Manifold-regularised Large-Margin $\ell_p$-SVDD for Multidimensional Time Series Anomaly Detection

  2025cites this paper
• Novelty detection on path space

  2025cites this paper
• Signature-Informed Selection Detection: A Novel Method for Multi-Locus Temporal Population Genetic Model with Recombination

  2025influential citation
• Kernel Learning for Mean-Variance Trading Strategies

  2025cites this paper
• pySigLib - Fast Signature-Based Computations on CPU and GPU

  2025cites this paper
• Global universal approximation with Brownian signatures

  2025cites this paper
• Random Controlled Differential Equations

  2025cites this paper
• Signature-Informed Transformer for Asset Allocation

  2025cites this paper
• Generalized Signature Method for Multivariate Time Series: A Data-Driven Framework for Feature Extraction

  2025cites this paper
• Signature Maximum Mean Discrepancy Two-Sample Statistical Tests

  2025influential citation
• Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models

  2025cites this paper
• Signature kernel ridge regression time series model: A novel approach for hydrological drought modeling using multi-station meteorological drought information

  2025cites this paper
• Discrete signature tensors for persistence landscapes

  2025cites this paper
• A User's Guide to KSig: GPU-Accelerated Computation of the Signature Kernel

  2025influential citation
• Scalable Signature Kernel Computations for Long Time Series via Local Neumann Series Expansions

  2025influential citation
• In-Context Learning of Stochastic Differential Equations with Foundation Inference Models

  2025cites this paper
• Numerical Schemes for Signature Kernels

  2025cites this paper
• Time Series Classification with Elasticity Using Augmented Path Signatures

  2025cites this paper
• Koopman-Equivariant Gaussian Processes

  2025cites this paper
• Expected Signature Kernels for L\'evy Rough Paths

  2025cites this paper
• Local regression on path spaces with signature metrics

  2025cites this paper
• Automotive industrial non-scalar data sharing

  2025cites this paper
• Signature Kernel Scoring Rule as Spatio-Temporal Diagnostic for Probabilistic Forecasting

  2025cites this paper
• On the signature of an image

  2024cites this paper
• Improving Handwritten Mathematical Expression Recognition via Similar Symbol Distinguishing

  2024cites this paper
• Nonparametric filtering, estimation and classification using neural jump ODEs

  2024cites this paper
• Average signature of geodesic paths in compact Lie groups

  2024cites this paper
• Scalable Signature-Based Distribution Regression via Reference Sets

  2024cites this paper
• Signature matrices of membranes

  2024cites this paper
• Transportation Market Rate Forecast Using Signature Transform

  2024cites this paper
• Generative modelling of financial time series with structured noise and MMD-based signature learning

  2024influential citation
• Topologies on unparameterised rough path space

  2024cites this paper
• Forecasting mortality rates with functional signatures

  2024cites this paper
• Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detection

  2024influential citation
• Neural Persistence Dynamics

  2024cites this paper
• Exact Gradients for Stochastic Spiking Neural Networks Driven by Rough Signals

  2024cites this paper
• Log-PDE Methods for Rough Signature Kernels

  2024cites this paper
• A Signature Based Approach towards Global Channel Charting with Ultra Low Complexity

  2024cites this paper
• A path-dependent PDE solver based on signature kernels

  2024cites this paper
• Signature Isolation Forest

  2024cites this paper
• Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes

  2024cites this paper
• Free probability, path developments and signature kernels as universal scaling limits

  2024influential citation
• Rough path theory

  2024cites this paper
• Signature Volatility Models: Pricing and Hedging with Fourier

  2024cites this paper
• Transportation Marketplace Rate Forecast Using Signature Transform

  2024cites this paper
• Improved climate time series forecasts by machine learning and statistical models coupled with signature method: A case study with El Niño

  2023cites this paper
• Signatures Meet Dynamic Programming: Generalizing Bellman Equations for Trajectory Following

  2023cites this paper
• FRUITS: feature extraction using iterated sums for time series classification

  2023cites this paper
• Random Fourier Signature Features

  2023influential citation
• SigFormer: Signature Transformers for Deep Hedging

  2023cites this paper
• Gaussian Processes Based Data Augmentation and Expected Signature for Time Series Classification

  2023cites this paper
• Shortest-path recovery from signature with an optimal control approach

  2023cites this paper
• A Bias-Variance-Covariance Decomposition of Kernel Scores for Generative Models

  2023cites this paper
• Global universal approximation of functional input maps on weighted spaces

  2023cites this paper
• Koopman Kernel Regression

  2023cites this paper
• Signature and Log-Signature for the Study of Empirical Distributions Generated with GANs

  2023cites this paper
• Signature Methods in Stochastic Portfolio Theory

  2023cites this paper
• S IGNATURE -I NFORMED S ELECTION D ETECTION : A N OVEL M ETHOD FOR M ULTI -L OCUS W RIGHT -F ISHER M ODELS WITH R ECOMBINATION

  2023influential citation
• The Signature Kernel

  2023influential citation
• The insertion method to invert the signature of a path

  2023cites this paper
• Infinite-dimensional reservoir computing

  2023cites this paper
• Neural signature kernels as infinite-width-depth-limits of controlled ResNets

  2023cites this paper
• Nyström M-Hilbert-Schmidt Independence Criterion

  2023cites this paper
• New Directions in the Applications of Rough Path Theory

  2023cites this paper
• Kernelized Cumulants: Beyond Kernel Mean Embeddings

  2023cites this paper
• Sig-Splines: universal approximation and convex calibration of time series generative models

  2023cites this paper
• TapTree: Process-Tree Based Host Behavior Modeling and Threat Detection Framework via Sequential Pattern Mining

  2023cites this paper
• Path signatures for diversity in probabilistic trajectory optimisation

  2023influential citation
• Eleven quick tips for data cleaning and feature engineering

  2022cites this paper
• Applications of Signature Methods to Market Anomaly Detection

  2022cites this paper
• Amortised Likelihood-free Inference for Expensive Time-series Simulators with Signatured Ratio Estimation

  2022influential citation
• Signature and Log-signature for the Study of Empirical Distributions Generated with GANs

  2022cites this paper
• Introduction to the signature method

  2022influential citation
• Learning with Signatures

  2022cites this paper
• Capturing Graphs with Hypo-Elliptic Diffusions

  2022influential citation
• Graph counting signatures and bicommutative Hopf algebras

  2022cites this paper
• Optimal Estimation of Generic Dynamics by Path-Dependent Neural Jump ODEs

  2022cites this paper
• Signature Methods in Machine Learning

  2022cites this paper
• Signature-Based Models: Theory and Calibration

  2022cites this paper
• Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models

  2022cites this paper
• Chaotic Hedging with Iterated Integrals and Neural Networks

  2022influential citation
• Two-dimensional signature of images and texture classification

  2022cites this paper
• A kernel Stein test of goodness of fit for sequential models

  2022cites this paper
• Two-parameter sums signatures and corresponding quasisymmetric functions

  2022cites this paper
• Approximate Bayesian Computation for Panel Data with Signature Maximum Mean Discrepancies

  2022cites this paper
• Reservoir kernels and Volterra series

  2022cites this paper
• Graph Profiles as Characters on Bicommutative Hopf Algebras

  2022cites this paper
• Maximum Mean Discrepancy on Exponential Windows for Online Change Detection

  2022cites this paper
• Framing RNN as a kernel method: A neural ODE approach

  2021influential citation
• Efficient and Accurate Gradients for Neural SDEs

  2021cites this paper
• How to extract knowledge from observed sequential data

  2021cites this paper
• Deep Signature Statistics for Likelihood-free Time-series Models

  2021cites this paper
• Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Series

  2021cites this paper