A new method compares unaligned datasets using log-Euclidean signatures of SPD matrices.
problem Efficiently comparing datasets with unknown alignment.
method Diffusion operators, Riemannian geometry, log-Euclidean metric.
result LES distance recovers meaningful structural differences, outperforming existing methods.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
A new mechanism for differentially private Fréchet mean on SPD matrices.
problem Privacy-preserving statistical summaries for SPD matrices.
method Tangent Gaussian mechanism for log-Euclidean metric.
result Significantly better utility and computational efficiency.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued representations often relies on Euclidean approximations or costly geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs.
result Generalizes batch normalization and multinomial logistic regression to broader classes of manifolds.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued data lacks reusable modules, specific network architectures, and efficient geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs for broad classes of Lie groups and gyrogroups.
result Generalizes batch normalization and multinomial logistic regression to Riemannian manifolds, including SPD and hyperbolic spaces.
Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …
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.
Defines knot signature invariant using G-signature theorem.
problem No specific problem stated; focuses on knot theory.
method Uses G-signature theorem to define knot invariant.
result Defines an invariant for strongly invertible knots.
Python package for SPD matrix distances, reproducible and extensible.
problem Computing distances between SPD matrices for various applications.
method Unified, extensible framework supporting multiple SPD metrics.
result Reproducible and accessible SPD matrix comparison tool.
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.
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…
Maximum Levine-Tristram signature of torus knots follows a reduction formula.
problem Determining the maximum Levine-Tristram signature for torus knots.
method Proved a reduction formula analogous to Gordon-Litherland-Murasugi's classical signature result.
result Maximum Levine-Tristram signature of torus knots satisfies a reduction formula.
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.
Introduces flat discrete signatures for financial data analysis.
problem Representing financial data for machine learning without continuous transformation.
method Introduced flat discrete signatures and discrete signatures, generalizing flat discrete signatures.
result Flat discrete signatures can represent quadratic variation relevant in finance.
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 …
We define the Analytical signature, the Hodge signature and the de Rham signature for a foliated manifold with boundary with foliation transverse to the boundary. We show that all these signatures coincide and a Hirzebruch formula is valid.
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.
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.
Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.
problem Computing signatures of torus links and their cores.
method Use Neumann's equivariant signatures and rewrite Hirzebruch's formula for torus links (without cores) in terms of integral points in a parallelogram.
result Rewritten Hirzebruch's formula for torus links with cores using integral points in a parallelogram.
This paper extends the C*-signature to non-Witt spaces using noncommutative geometric methods.
problem Extending the signature to non-Witt spaces with noncommutative geometric methods.
method Noncommutative geometric methods, combinatorial framework, and comparison with analytical signature.
result Constructing the C*-signature on non-Witt spaces.
Paper generalizes path signature using fractional calculus for improved machine learning.
problem Improving path signature for machine learning applications.
method Introduces two new signatures inspired by fractional calculus and machine learning considerations.
result Significant accuracy improvements in handwritten digit recognition.
We present a novel method for extracting cancer signatures by applying statistical risk models (http://ssrn.com/abstract=2732453) from quantitative finance to cancer genome data. Using 1389 whole genome sequenced samples from 14 cancers, we identify an "overall" mode of somatic mutational noise. We give a prescription …
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.
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.
New method calculates signatures of biquotients.
problem Signature calculation for homogeneous spaces.
method Generalization of Hirzebruch's computation to biquotients.
result Signature of biquotients computed for equal rank cases.
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.
Link signature limit depends on linking matrix under specific polynomial condition.
problem Limits of Tristam-Levine signature function under precise polynomial conditions.
method Analysis of Alexander polynomial and linking matrix.
result Limit of Tristam-Levine signature at 1 determined by linking matrix under specific polynomial condition.
Path signatures improve hedging of exotic derivatives in non-Markovian models.
problem Hedging exotic derivatives under non-Markovian stochastic volatility models.
method Investigates path signatures in deep and shallow learning contexts, comparing neural networks and regression approaches.
result Path signatures outperform LSTM in most cases and yield more accurate results in hedging.
Develops a new family of signature-changing models on metric manifolds.
problem Signature changes in metric manifolds.
method One-parameter family of Lorentz-Riemann models, local expressions around change.
result Generalizes existing signature-changing models.
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…
Paper confirms Kashaev's signature conjecture for links.
problem Proving Kashaev's conjecture about link invariants.
method Using Seifert surface definition and diagrammatic approach.
result Established Kashaev's conjecture, providing a new formula for Alexander polynomial.
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 finds infinitely many surface bundle types with zero signature.
problem Characterizing homeomorphism types of surface bundles with specific signatures.
method Analyzing atoroidal surface bundles over surfaces.
result Infinitely many homeomorphism types of atoroidal surface bundles over surfaces with signature zero.
In this note, we lay the groundwork for a new approach to the problem of group-signature classification of group actions on closed Riemann surfaces. This new approach first focuses on analyzing the low level arithmetic conditions on signatures before invoking the more complicated group theory. We provide the complete f…
A new graph signature invariant to graph automorphisms.
problem Graph symmetry and feature generation.
method Power spectrum signature derived from squared graph Fourier transform.
result Power spectrum signature is stable under graph perturbations.
The study explains why signature methods work in commodity futures term structure classification.
problem Lack of interpretability in signature methods for term structure classification.
method Introducing signature perturbations to explain the success of signature-based classification.
result The volatility of the convenience yield is the major discriminant for commodity markets classification.
Study on signatures of positive braids with bounds derived.
problem Understanding signatures of positive braids and their invariants.
method Derived lower bounds for Levine-Tristram signatures, and upper and lower bounds on signature ratios.
result Established bounds on signatures of positive braids, uniformly valid across monoids.
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…
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 define and study the signature, A-hat genus and higher signatures of the quotient space of an S1-action on a closed oriented manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.
Motivation : Molecular signatures for diagnosis or prognosis estimated from large-scale gene expression data often lack robustness and stability, rendering their biological interpretation challenging. Increasing the signature's interpretability and stability across perturbations of a given dataset and, if possible, acr…
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.
Normal distribution found for 2-bridge knots signatures.
problem Distribution of signatures for 2-bridge knots.
method Calculated average signature, introduced s(c,σ), used limit theorem. result Distribution of signatures approaches normal as knot complexity increases.
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.
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.
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 …
Construct symplectic Lefschetz fibrations with any signature and spin type.
problem Existence of symplectic Lefschetz fibrations with arbitrary signature.
method Develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature and spin type.
result Solve a long-standing conjecture on the existence of such fibrations with positive signature.
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.