We provide an introduction to the signature method, focusing on its theoretical properties and machine learning applications. Our presentation is divided into two parts. In the first part, we present the definition and fundamental properties of the signature of a path. The signature is a sequence of numbers associated …
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48 results for “property signatures”
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.
A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifol…
The paper classifies links based on signature and crossing number properties.
problem Understanding the relationship between signature and crossing number of knots and links.
method Refined theorems and comprehensive classification of links with specific properties.
result Identification of all links where signature + crossing number = 2, closures of positive 3-braids.
The paper revisits expected signatures in semimartingale models, providing new formulae and simplifying complexity.
problem Computing expected signatures in semimartingale models.
method Revisits and provides new formulae for computing expected signatures in a general semimartingale setting.
result Log-transform of expected signatures simplifies complexity, leading to signature cumulants.
We study properties of the signature function of the torus knot . First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.
The Signature Kernelmath.PR
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.
Average signature measures geodesics in Lie groups.
problem Understanding geometric properties of Lie groups through geodesic paths.
method Introducing average signature and using it with trace operation to recover geometric properties.
result Average signature can recover geometric properties like dimension, diameter, volume, and scalar curvature.
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.
The paper examines the consistency of Lasso regression applied to signature analysis of time series data.
problem Consistency of Lasso regression in signature analysis of time series data.
method The paper studies the consistency of Lasso regression applied to signature analysis of time series data, both theoretically and numerically.
result The Lasso regression is consistent both asymptotically and in finite sample for certain types of time series and processes.
This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology …
Local signature-based expansionsq-fin.MF
Study local expansions of continuous-time processes using Ito signature properties.
problem Analyzing local expansions of continuous-time processes and their moments.
method Using the Ito signature, a basis of iterated integrals, to conduct expansions of the process' characteristic function.
result Explicit coefficients and stochastic representations for asymptotics as time shrinks or diverges.
This note shows every integer can be a signature of a hyperbolic 4-manifold.
problem Finding signatures of hyperbolic 4-manifolds.
method Using Long and Reid's theorem and explicit construction of a 24-cell manifold.
result Every integer is the signature of a non-compact, oriented, hyperbolic 4-manifold.
A recent line of work has uncovered a new form of data poisoning: so-called \emph{backdoor} attacks. These attacks are particularly dangerous because they do not affect a network's behavior on typical, benign data. Rather, the network only deviates from its expected output when triggered by a perturbation planted by an…
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.
In this paper, by combining modularity of the Witten genus and the modular forms constructed by Liu and Wang, we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds.
In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4-manifold with even intersection form. Furthermore, we introduce a new geometric group invariant and discuss some of its properties.
New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordi…
Universal approximation for rough paths and Lévy processes.
problem Approximating continuous functionals of càdlàg paths.
method Linear functionals of time-extended signatures.
result Universal approximation theorem for continuous functionals of càdlàg paths.
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
problem Extending graph signatures to Klein graphs and foams.
method Developed an analogy of Murasugi's bounds and used signatures to lower bound knot properties.
result Lower bounds on negative orbifold Euler characteristics and unknotting numbers.
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.
Researchers develop Malliavin calculus for signatures, simplifying option Greeks computation.
problem Lack of tractability and explicit representations in Malliavin calculus.
method Focus on finite linear combinations of time-extended Brownian motion signatures, derive explicit formulas for Malliavin derivative, and compute Greeks for path-dependent options.
result Closed-form expressions for classical operators of Malliavin calculus, providing algebraic formulations.
Expected signatures map data streams to lower dimensions, improving ML performance.
problem Leveraging model-free embeddings for domain-agnostic machine learning.
method Expected signatures map data streams to lower dimensions, with convergence results bridging empirical and theoretical estimators.
result A modified expected signature estimator with lower mean squared error for martingale processes.
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.
Extended signatures help distinguish non-concordant links.
problem Distinguishing non-concordant links using signatures.
method Defined and studied an n-variable extension of the Levine-Tristram signature, proving it a concordance invariant on a dense subset of the torus.
result Found an infinite family of 3-component links not concordant to their mirror images, detectable only by the extended signature.
Path signatures adapted for Lie groups improve action recognition in computer vision.
problem Improving action recognition in computer vision with geometric constraints.
method Lifting path signatures to Lie groups and proving universality and characteristic property.
result Path signatures on Lie groups provide comparable performance to shallow learning approaches in action recognition.
This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot and a representation of the knot group, we define a twisted signature function . This invariant satisfies many of the same algebraic pr…
The paper introduces surface signatures for irregular surfaces and rough surfaces.
problem Characterizing and integrating highly irregular paths and surfaces.
method Introducing surface signatures and proving extension theorems.
result Surface signatures are universal for surface holonomy and rough surfaces.
In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable p…
Researchers extend geodesic orbit properties to pseudo-Riemannian nilmanifolds of specific signature.
problem Extending geodesic orbit properties to pseudo-Riemannian manifolds of signature (n-2,2).
method Analyzing trans-Lorentz nilmanifolds with reductive decomposition and nilpotent subgroup.
result Characterization of nilpotent subgroups and structure of nilmanifolds.
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.
Clusters of crypto assets by path signature improve diversification and reduce fees.
problem Building diversified portfolios of volatile cryptocurrencies.
method Clustering digital assets using path signatures to identify similar behavior patterns.
result Optimal portfolios outperform unfiltered ones, reducing transaction fees.
New method solves optimal stopping problems using rough path signatures.
problem Optimal stopping problems in finance and other fields.
method Using rough path signatures and deep neural networks.
result Solves optimal stopping problems efficiently under minimal assumptions.
Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miscenko for manifolds. Our construction has all of the expected pr…
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.
We are studying the harmonic and twistor equation on Lorentzian surfaces, that is a two dimensional orientable manifold with a metric of signature . 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…
Sig-Splines model uses signatures and splines for time series data, achieving universality and convexity.
problem Creating a generative model for multivariate time series data.
method Combines linear transformations and signature transforms into a neural spline flow.
result Achieves universality and introduces convexity in model parameters.
The space writhes and signatures of polymer knotscond-mat.soft
Study shows space writhe closely correlates with knot signature in polymers.
problem Understanding the relationship between space writhe and knot signatures in knotted polymers.
method Performed Langevin dynamics simulations of knotted polymers to measure space writhe.
result Space writhe is strongly correlated with knot signature in complex knots.
Werner Meyer constructed a cocycle in which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this pap…
Explicit formulas for Hattori-Stong integrability conditions and manifolds' signature properties.
problem Understanding and calculating the signature of stably almost-complex manifolds.
method Combinatorial techniques to derive explicit formulas for Chern class coefficients and signature conditions.
result An evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers.
The paper studies Gauss sums and their applications in algebra and topology.
problem Computing signatures of bilinear forms and Poincaré spaces.
method Investigates properties of Gauss sums and applies them to algebraic and topological problems.
result Reproves and generalizes results on signatures mod 8 of bilinear and Poincaré spaces.
While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses:…
In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Fina…
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.
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.
New complexity measure for shake-slice knots established.
problem Defining and measuring complexity for shake-slice knots.
method Using dualizable patterns and studying knot signatures.
result Existence of -shake-slice knots with specified complexity.
Study hyperplanes in abelian groups and their signatures for manifold identification.
problem Identifying manifolds based on their homology groups and coordinate hyperplanes.
method Investigates isomorphisms preserving coordinate hyperplanes in products of cyclic groups.
result Recovering coordinate hyperplanes from their union and applying to manifold identification.