Signature kernel handles sequential data with theoretical and practical advantages.
problem Handling sequential data efficiently and accurately.
method Positive definite kernel based on stochastic analysis with efficient computation.
result Strong empirical performance and theoretical guarantees.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
pySigLib speeds up signature-based computations on CPUs and GPUs.
problem Efficient signature-based computations on large datasets and long sequences.
method Optimised Python library for CPU and GPU, novel differentiation scheme.
result Accurate gradients at a fraction of the runtime of existing libraries.
New methods price American options in rough volatility models.
problem Pricing American options under rough volatility.
method Integrating deep-signature and signature-kernel learning into optimal stopping problem solutions.
result Performance comparison in rough Heston and rough Bergomi models.
Develops a new solver for path-dependent PDEs using signature kernels.
problem Solving path-dependent PDEs (PPDEs) efficiently and accurately.
method Uses signature kernels to solve PPDEs by approximating the solution with minimal norm in a reproducing kernel Hilbert space.
result Proves the consistency of the numerical scheme, ensuring convergence to PPDE solutions as the number of collocation points increases.
A Python package for GPU-accelerated signature kernel computation.
problem Efficient computation of signature kernels for sequential data.
method GPU-accelerated algorithms and tensor sketches.
result New algorithm outperforms existing methods.
Efficiently computes sparse signature coefficients using kernels.
problem Lack of efficient methods for sparse signature coefficients.
method Signature kernels and PDE-based methods.
result Sparse groups of signature coefficients can be isolated effectively.
Estimates path-valued data using signature metrics and local kernels.
problem Nonparametric regression and classification for path-valued data.
method Combines signature transform and local kernel regression.
result Establishes convergence bounds and demonstrates competitive accuracy.
Accelerates signature kernel computation for sequences.
problem Severe computational bottleneck in computing signature kernel.
method Random Fourier features to accelerate signature kernel computation.
result Uniform approximation guarantees for unbiased estimator with linear computation time.
Signature kernel scoring rule improves weather forecasting by capturing temporal and spatial dependencies.
problem Lack of suitable scoring rules for probabilistic weather forecasting.
method Reframe weather variables as continuous paths using iterated integrals (signature kernels) to capture temporal and spatial dependencies.
result Signature kernel scoring rule outperforms conventional methods in weather forecasting, especially for long-term forecasts.
New algorithms compute Volterra signature efficiently for time series analysis.
problem Efficient computation of Volterra signature with matrix-valued kernels.
method Decomposed Chen-type convolution relation, introduced FFT-based and exact recursion algorithms.
result Efficient algorithms for Volterra signature computation with various complexities.
Scalable machine learning with path signatures for time series and graphs.
problem Challenges in real-world time series and graph data.
method Combines rough path theory with probabilistic, deep, and kernel methods.
result Scalable models for time series and graph data.
New method uses path signatures for efficient likelihood estimation in time-series data.
problem Intractable likelihood functions in complex dynamic models.
method Kernel classifier based on path signatures for sequential data.
result Path signatures yield highly performant classifiers, even with low sample numbers.
A new algorithm for high-dimensional hedging problems.
problem High-dimensional, path-dependent hedging problems.
method Signature-based algorithm using operator-valued kernels and geometric rough paths.
result Theoretical guarantees on existence and uniqueness of a global minimum.
New metrics improve probabilistic forecasting, especially for rare events.
problem Current evaluation frameworks for probabilistic forecasting assume independence and lack sensitivity to tail events.
method Proposed signature kernel-based metrics: Sig-MMD and CSig-MMD.
result These metrics capture complex dependencies and prioritize tail event prediction.
RNNs are reinterpreted as kernel methods using neural ODEs.
problem Improving generalization and stability of RNNs.
method Connecting RNNs to neural ODEs and reproducing kernel Hilbert spaces.
result RNNs can be viewed as linear functions of a specific feature set.
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 proce…
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…
Framework combines random features with CDEs for efficient time-series learning.
problem Efficient training of time-series models with strong inductive bias.
method Random Fourier CDEs and Random Rough DEs using continuous-time reservoirs and log-ODE discretization.
result Unified perspective on random-feature reservoirs and path-signature theory.
This paper develops a path-first theory using signatures and jump lifts for self-exiting processes.
problem Developing a universal coordinate system for various types of paths and processes.
method Using signatures, jump lifts, and expected signatures, the paper presents a geometricity framework with algebraic properties and obstructions.
result The framework links various mathematical concepts and offers four main contributions to understanding and modeling self-exiting processes.
Volterra signature provides a clear, interpretable feature for history-dependent systems.
problem Learning from non-Markovian time series with implicit memory mechanisms.
method Develops Volterra signature as a tensor algebra representation weighted by a temporal kernel, proving injectivity and universal approximation.
result Volterra signature leads to linear functionals and universal approximation, improving dynamic learning tasks.
Bayesian time series forecasting improves by dynamically adapting to recent information.
problem Lack of forgetting mechanism in signature kernel for time series forecasting.
method Introducing a novel forgetting mechanism for signature features using Random Fourier Decayed Signature Features (RFDSF) with Gaussian processes (GPs).
result Demonstrates superior performance compared to other GP-based alternatives and state-of-the-art probabilistic time series forecasting algorithms.
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
Paper introduces non-adversarial training for Neural SDEs using signature kernel scores.
problem Stability and mode collapse issues in adversarial training of Neural SDEs.
method Uses signature kernel scores as objective function for non-adversarial training.
result Non-adversarial training leads to better performance and more stable models.
sig-MMD tests compare path distributions using kernel methods.
problem Comparing path distributions in stochastic processes.
method Signature kernel for path space valued distributions.
result sig-MMD can lead to Type 2 errors in limited data settings.
Develops new techniques for learning from sequential data groups.
problem Learning from groups of inputs rather than individual inputs.
method Introduces feature-based and kernel-based learning techniques for sequential data.
result Achieves state-of-the-art performance on various real-world examples.
Develops a kernel-based framework for dynamic trading strategies.
problem Optimizing portfolios with temporal dependencies in asset dynamics.
method Parameterizes trading strategies as functions in RKHS, enabling flexible, non-Markovian approaches.
result Significantly outperforms classical Markovian methods in synthetic and market-data examples.
Kernel smoothing on unknown manifolds with bounds and asymptotic normality.
problem Data on unknown manifolds without boundaries.
method Finite sample bounds and asymptotic normality for kernel smoothing and its derivatives.
result Established finite sample bounds and asymptotic normality for kernel smoothing.
Study describes signatures of Ricci curvature on nilmanifolds.
problem Understanding Ricci curvature on nilpotent Lie groups.
method Link between Ricci endomorphism kernel and closed orbits in representation of GL group, proved using real GIT.
result Complete description of Ricci curvature signatures on nilmanifolds.
A hybrid framework for American option pricing under time-varying rough volatility.
problem Pricing American options under time-varying rough volatility.
method Signature method combined with gradient-boosted ensemble for Hurst parameter estimation, regime switch, and Random Fourier Features for acceleration.
result The proposed hybrid framework improves performance over fixed-roughness baselines and reduces duality gaps in some regimes.
Study delocalized eta invariants for signature operators on proper manifolds.
problem Define and analyze delocalized eta invariants for signature operators on proper manifolds.
method Develop detailed heat-kernel analysis and apply to proper manifolds with boundary.
result Prove index formulas relating delocalized eta invariants to Atiyah-Patodi-Singer indices.
The paper develops CI tests for causal discovery in SDEs.
problem Inferring causal structure from stochastic dynamical systems.
method Developed CI constraints and a CI test for SDEs.
result Proposed CI test outperforms existing methods.
We develop a Bayesian approach to learning from sequential data by using Gaussian processes (GPs) with so-called signature kernels as covariance functions. This allows to make sequences of different length comparable and to rely on strong theoretical results from stochastic analysis. Signatures capture sequential struc…
Paper introduces FDM for efficient training of Neural SDEs.
problem Training Neural SDEs using existing methods is computationally expensive and unstable.
method Developed a novel scoring rule called Finite Dimensional Matching (FDM) to bypass signature kernels and reduce training complexity.
result FDM achieves superior performance in terms of computational efficiency and generative quality.
SigGPDE scales sparse Gaussian processes for sequential data.
problem Predicting and quantifying uncertainty in sequential data.
method Sparse variational inference framework for Gaussian Processes, leveraging GP signature kernel gradients as PDE solutions.
result Significant computational gains and state-of-the-art performance on large sequential datasets.
We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Pato…
Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.
problem Analyzing Dirac operators on complex geometric spaces.
method Construct heat kernel, prove self-adjointness and Fredholm properties, establish index formula.
result Proved Dirac operators are essentially self-adjoint and Fredholm.
Generative model for financial time series using structured noise and signature learning.
problem Creating synthetic financial data to reflect real-world market dynamics.
method Structured noise, moving average model, signature transform, reinforcement learning.
result Model effectively captures key financial characteristics and outperforms existing methods.
The abstract discusses embedding theorems for pseudo-Kähler manifolds.
problem Embedding theorems for pseudo-Kähler manifolds.
method Using quantizable pseudo-Kähler manifolds and Hermitian line bundles, the asymptotic expansion of Bergman kernels is analyzed.
result The asymptotic expansion of Bergman kernels implies analogues of Kodaira embedding theorem and Tian's almost-isometry theorem.
Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Mach…
Let φ∈C∞(Cn) be a given real valued function. We assume that $\pr\ddbarφ$ is non-degenerate of constant signature (n−,n+) on Cn. When q=n−, it is well-known that the Bergman kernel for (0,q) forms with respect to the k-th weight e−2kφ, k>0, admits a full asymptotic expansi…
We analyze in this paper a random feature map based on a theory of invariance I-theory introduced recently. More specifically, a group invariant signal signature is obtained through cumulative distributions of group transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as …
Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
Distinguishing between classes of time series sampled from dynamic systems is a common challenge in systems and control engineering, for example in the context of health monitoring, fault detection, and quality control. The challenge is increased when no underlying model of a system is known, measurement noise is prese…
Framework for training stochastic spiking neural networks with rough signals.
problem Training stochastic spiking neural networks with noisy spike timing and dynamics.
method Rough path theory and signature kernels for gradient computation.
result Pathwise gradients of SSNNs' trajectories and event times exist and satisfy a recursive relation.
Scalable GPLVM reduces complexity in scRNA-seq data, accounting for technical and biological confounders.
problem Complexity and confounders in scRNA-seq data hamper interpretation.
method Extended Gaussian process latent variable model (GPLVM) to handle large datasets.
result Framework reconstructs latent signatures and captures disease-specific gene expression.
Informative and discriminative feature descriptors play a fundamental role in deformable shape analysis. For example, they have been successfully employed in correspondence, registration, and retrieval tasks. In the recent years, significant attention has been devoted to descriptors obtained from the spectral decomposi…
In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator B in the sense of Se…