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.
The paper adapts differential signatures to algebraic curves under group actions.
problem Equivalence problem for complex plane algebraic curves under group actions.
method Adapting differential signature construction to algebraic curves, using classifying invariants.
result Explicit sets of rational classifying invariants and formulas for signature curve degree.
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.
Generalizes quantum integrability to all signatures for projectively equivalent metrics.
problem Quantum integrability for Beltrami-Laplace operators across various signatures.
method Shows that Killing tensors constructed from projectively equivalent metrics correspond to commuting differential operators.
result Quantum integrability for Beltrami-Laplace operators is established for all signatures.
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.
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.
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.
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 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.
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.
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.
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. New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
For a normal covering over a closed oriented topological manifold we give a proof of the L2-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C^*_max-version of the Bau…
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.
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.
Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by 4. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent…
New findings on non-congruent curves with identical signatures.
problem Identifying congruence of non-degenerate curves with non-simple signatures.
method Associated directed graphs to signatures and used paths to reflect global and local symmetries.
result Non-congruent, non-degenerate curves can have identical signatures.
We compute a closed formula for the class of the closure of the locus of curves in Mg that admit an abelian differential of signature κ=(k1,...,kg−2).
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.
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.
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 …
Optimizes wavelets for graph classification using spectral wavelet signatures and persistence diagrams.
problem Graph classification with geometric properties encoded in persistence diagrams.
method Optimizes spectral wavelets for graph datasets to capture best-suited features for classification.
result Competitive performance in graph classification problems compared to other persistence-based architectures.
In this paper, we adapt part of Weinberger, Xie and Yu's breakthrough work, to define additive higher rho invariant for topological structure group by differential geometric version of signature operators, or in other words, unbounded Hilbert-Poincaré complexes.
We give explicit formulas for the intertwinors on the differential form bundles over Sp−1×Sq−1 with the standard pseudo-Riemannian metric g=−gSp−1+gSq−1 of signature (p−1,q−1). As a special case, we construct conformally invariant differential operators of all even orders.
We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle E, endowed with a bundle metric of split signature and further datum. The geometric structure is associated with a so-called generalised G-structure and characterised by an E-spinor ρ, which we can regard as a …
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.
On conformal manifolds of even dimension n≥4 we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new complexes is elliptic if the signature is Riemannian. We also construct gauge compani…
Study of pure spinors on neutral manifolds with applications to supersymmetric solutions.
problem Characterizing pure spinors and their properties on neutral manifolds.
method Using the theory of real spinorial forms and differential systems, the square of pure spinors is analyzed.
result Non-pure spinors correspond to specific structures in signature (4,4), and parallel spinors are characterized by differential systems.
We show how to compute the spectral flow of the odd signature operator ±∗dat−dat∗ along an analytic path of flat connections at on a bundle over a closed odd-dimensional manifold in terms of Massey products in the DGLA of bundle-valued differential forms. To obtain this information, we set up a sequence…
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,…
Machine learning identifies 3-manifold triangulations using isomorphism signatures.
problem Differentiating and classifying 3-manifolds and their Dehn surgeries.
method Training machine learning models on isomorphism signatures derived from 3-manifold triangulations and Pachner graphs.
result Gradient saliency analysis reveals key parts of the language-like encoding scheme.
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.
Signatory calculates signature and logsignature transforms efficiently on CPU and GPU.
problem Efficient computation of signature and logsignature transforms for machine learning.
method CPU and GPU parallelism, backpropagation, efficient precomputation strategies, algorithmic improvements.
result Substantial speedups on CPU and GPU, including real-world applications.
For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may…
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.
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.
Study of differential spinors on three-manifolds with skew-torsion.
problem Characterizing differential spinors on Lorentzian three-manifolds with skew-torsion.
method Developed spinorial polyforms and used them to study differential spinors, proving that every differential spinor is equivalent to an isotropic line preserved by a metric connection with skew-torsion.
result Obtained structural results about Lorentzian three-manifolds equipped with skew-torsion parallel spinors, which are necessarily Kundt and geodesically complete in the compact case.
For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,E,H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H=∑ij+1H2j+1 is an odd-degree closed differential form on X and H2j+1 is a real-valued differential form of degree…
We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS sa…
Parallel spinors help characterize G2* structures and isotropic forms.
problem Characterizing G2* structures and isotropic forms on pseudo-Riemannian manifolds.
method Using a correspondence between irreducible parallel spinors and solutions of a differential system for three-forms.
result Explicit description of isotropic irreducible spinors in signature (4,3) and characterization of G2* structures.