Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
We study the self-dual Yang-Mills equations in split signature. We give a special solution, called the basic split instanton, and describe the ADHM construction in the split signature. Moreover a split version of t'Hooft ansatz is described.
Deep signature/log-signature FBSDE algorithm improves accuracy and training time.
problem Solving FBSDEs with state and path dependent features.
method Incorporates deep signature/log-signature transformation into RNN model.
result Improves accuracy and training time compared to existing methods.
Global approximation for piecewise linear paths via signatures.
problem Global approximation theorems for piecewise linear paths.
method Using signatures of piecewise linear paths and their density in Lp-norms. result Linear functionals of signatures are dense in Lp-norms under an integrability condition. Signature tensors uniquely identify ODE solutions.
problem Identifying ODE solutions from signature tensors.
method Geometric theory of nonlinear systems of ODEs.
result Necessary and sufficient algebraic conditions for signature tensors to represent ODE solutions.
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.
The paper develops a deep signature approach for option pricing under non-Markovian stochastic volatility models.
problem Pricing options under non-Markovian stochastic volatility models is challenging due to the dependence on historical paths.
method Reformulate the asset dynamics as a rough stochastic differential equation and represent rough paths via signatures. Apply standard analytical tools to solve the transformed equation.
result The deep signature approach provides a theoretically grounded and computationally efficient framework for option pricing.
Novel signature approach for pricing and hedging path-dependent options with market frictions.
problem Pricing and hedging path-dependent options with market frictions.
method Signature approach, mean-quadratic variation criterion, non-standard infinite-dimensional Riccati equations, time-augmented signature, non-Markovian stochastic control problem.
result Effective hedging strategies in frictional markets with low-truncated signature approximations.
New method uses randomised signatures for generating financial time series data.
problem Generating synthetic financial time series data accurately.
method Introduced a Wasserstein-type distance based on discrete-time randomised signatures.
result Demonstrated universal approximation for randomised signatures on continuous functions.
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.
We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applicat…
Positive braids have a signature bound by their Betti number.
problem Bounding the signature of positive braids.
method Using the first Betti number as a lower bound for the signature.
result The signature is bounded from below by one-quarter of the first Betti number.
Study on martingale property and moment explosions in signature volatility models.
problem Analyzing the martingale property and moment explosions in signature volatility models.
method Fine analysis of the explosion time of a signature stochastic differential equation.
result The price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative.
New criteria ensure uniqueness of curve signatures, robust to metric variations.
problem Ensuring uniqueness of curve signatures in differential geometry.
method Introducing new methods through differential equations and higher order derivatives.
result New criteria for curve signature uniqueness in general settings.
Paper combines RNN and signatures for learning functions on streamed multimodal data.
problem Learning functions on streamed multimodal data.
method Hybrid Logsig-RNN algorithm combining signatures and RNN.
result Hybrid algorithm achieves outstanding accuracy with superior efficiency and robustness.
Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Universal approximation for stochastic processes using Brownian motion.
problem Approximating stochastic processes with linear functionals.
method Establishing Lp-type universal approximation theorems for rough path spaces. result Linear functionals on the signature of time-extended Brownian motion can approximate any p-integrable stochastic process. Study classifies half conformally flat GQE manifolds of signature (2,2).
problem Classifying half conformally flat generalized quasi-Einstein manifolds.
method Analysis and examples provided.
result Natural affine quasi-Einstein equation derived.
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.
New SDEs from affine and polynomial perspectives for path-dependent processes.
problem Characterizing path-dependent stochastic processes.
method Affine and polynomial processes, signature SDEs, Fourier-Laplace transform, Riccati and linear ODEs.
result Explicit formulas for the Fourier-Laplace transform and expected values of entire functions of signature processes.
Sig-DEG speeds up diffusion models by distilling them into faster approximations.
problem Computational intensity of diffusion models at inference time.
method Signature-based differential equation generation to summarize Brownian motion.
result Sig-DEG reduces inference steps by an order of magnitude while maintaining generation quality.
Paper introduces branched signature model for efficient computation and data-driven applications.
problem Efficient computation and data-driven modeling of branched rough paths.
method Develops a universal approximation theorem and constructs an extension map to realize branched signatures.
result Explicit construction of branched signatures via an extension map for efficient computation.
Unified and simplified signature method for multivariate time series.
problem Challenging application of signature method due to its flexibility.
method Generalised signature method unifying various techniques.
result Competitive performance against benchmarks for multivariate time series classification.
Novel model for predicting event intensities from static and time series data.
problem Predicting event intensities from static and irregularly sampled time series data.
method Neural controlled differential equations and signature-based CoxSig model.
result The CoxSig model provides theoretical learning guarantees and performs well on various datasets.
We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore,…
The study classifies Android malware using minhashing and Structural Equation Models.
problem Lack of consensus and consistency in malware signatures across different antivirus engines.
method Analyzed over 250k malware signatures from 61 engines on 82k Android apps.
result Identified 41 malware classes grouped into Adware, Harmful Threats, and Unknown categories.
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.
The paper defines signatures for Witt spaces with boundary and proves their equality.
problem Defining and proving signatures for Witt spaces with boundary.
method Introducing de Rham and Hodge signatures, extending index theory, and using von Neumann algebras.
result Equality of de Rham and Hodge signatures on Witt spaces with boundary.
Study of Schrödinger equation for particles on curved surfaces.
problem Modeling particles constrained to curved surfaces in Minkowski space.
method Developed a new Schrödinger equation for surfaces in R13 and analyzed its properties. result Found a geometry-induced potential with signature-dependent terms.
We are studying the harmonic and twistor equation on Lorentzian surfaces, that is a two dimensional orientable manifold with a metric of signature (1,1). We will investigate the properties of the solutions of these equations and try to relate the conformal invariant dimension of the space of harmonic and twistor spin…
Introduces Exponentially Weighted Signature for better path representation.
problem Uniform treatment of historical information in signatures.
method Generalizes EFM signature to bounded linear operators, enabling contextualised temporal weighting.
result EWS is the unique solution to a linear controlled differential equation and generalizes state-space models.
Study on nilpotent Lie algebras with specific metrics.
problem Classifying nilsolitons in nilpotent Lie algebras.
method Classification up to dimension 9, reduction to linear and polynomial equations.
result Complete classification of nice nilsolitons in various dimensions and signatures.
This review discusses solutions to Einstein's equations using twistor theory.
problem Finding solutions to Einstein's vacuum equations using twistor theory.
method Holomorphic vector bundles on twistor space and patching matrices.
result Holomorphic patching matrix P is simpler than the metric and determines the rod structure. Study explores Nahm-Schmid equations and their connection to hypersymplectic geometry.
problem Exploring Nahm-Schmid equations and their integrable nature.
method Dimensional reduction, explicit solutions, Lax-pair formulation, conservation laws, spectral curves, and hypersymplectic geometry.
result Established the connection between Nahm-Schmid equations and hypersymplectic geometry.
Develops Palatini formalism for pseudo-Finsler metrics, recovering classical results.
problem Developing a formalism for pseudo-Finsler metrics of any signature.
method Substituting scalar curvature with Finslerian Ricci scalar in Einstein-Hilbert-Palatini functional.
result Recovery of classical results in Lorentzian signature with vanishing mean Landsberg tensor.
New types of Einstein manifolds discovered from homogeneous surfaces.
problem Understanding quasi-Einstein equations on homogeneous surfaces.
method Modified Riemannian extension and warped product construction.
result Discovery of new Einstein manifolds.
Considering a non-constant smooth solution f of the Tanno equation on a closed, connected Kähler manifold (M,g,J) with positively definite metric g, Tanno showed that the manifold can be finitely covered by $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS})$, where gFS denotes the Fubini-Study metric of constant hol…
SigMA uses signatures and attention to estimate parameters in fBm-driven SDEs.
problem Estimating parameters in SDEs driven by fBm is challenging due to non-Markovian and semimartingale issues.
method SigMA integrates path signatures with multi-head self-attention, using convolutional and MLP layers.
result SigMA outperforms other methods in accuracy, robustness, and model compactness.
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.
Signature volatility models are analyzed for existence, arbitrage, completeness, and hedging-error decomposition.
problem Existence, arbitrage, completeness, and hedging-error decomposition of signature volatility models.
method Global existence and uniqueness of strong solutions, asset-pricing, market completeness, and hedging-error decomposition derived through structural results.
result Signature volatility models are structurally sound with existence, arbitrage, completeness, and hedging-error decomposition.
Abstract properties of hypersurface data analyzed in spherical symmetry.
problem Analyzing hypersurface data in spherical symmetry.
method Study of hypersurface data properties, gauge group, and curvature tensor.
result General solution of Einstein field equations in vacuum and Lorentzian ambient signature.
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …
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.
Neural SDEs improve time series generation efficiency.
problem High memory and computational costs in GANs for time series.
method Conditional Neural Stochastic Differential Equations (SDEs).
result More memory efficient and faster than traditional methods.
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.
New concept of Lorentzian-Euclidean black holes and metric transitions explored.
problem Signature-changing spacetimes and their geometric properties.
method Introduction and analysis of Lorentzian-Euclidean black holes and transitions.
result Consistency of proper time to horizon in Lorentzian-Euclidean black holes.
New method uses models from regularity structures as features in machine learning.
problem Learning solutions to PDEs with low regularity.
method Developed a flexible definition of model feature vectors and two algorithms for combining them with linear regression.
result Advantage in learning solutions to PDEs compared to alternative methods.
We demonstrate how the complex integral formula for the Airy functions arises from Penrose's twistor contour integral formula. We then use the Lax formulation of the isomonodromy problem with one irregular singularity of order four to show that the Airy equation arises from the anti-self-duality equations for conformal…