Paper evaluates squared-exponential covariance function for Gaussian processes with integral observations.
problem Evaluating double line integrals of the squared exponential covariance function in Gaussian processes.
method Proposes a new approach to reduce double integrals to a single integral using the error function and efficiently computed with numerical techniques.
result Shows superior numerical robustness and accuracy compared to existing methods.
We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated inte…
Study shows Euler discretizations preserve exponential integrability of CIR process.
problem Analyzing exponential integrability of CIR process and its discretizations.
method Examined various Euler discretizations with truncation and reflection at 0.
result Preservation of exponential integrability for implicit and explicit Euler-Maruyama discretizations.
Tensor networks improve integration accuracy for high-dimensional problems.
problem Integration of high-dimensional functions with exponential convergence.
method Regression-free tensor network representations for integration.
result Exponential convergence achieved for non-analytic integrands.
We represent the coordinate ring of algebraic hulls (which are generalizations of the Malcev completions of nilpotent groups for solvable groups) of solvmanifolds G/Γ by using Miller's exponential iterated integrals (which are extensions of Chen's iterated integrals) of invariant differential forms.
Improved KLMC for sampling under various conditions.
problem Stable simulation of kinetic Langevin dynamics under different parameters.
method Revisited synchronous Wasserstein coupling analysis with stochastic exponential Euler discretization.
result Exponential integrator can simulate kinetic Langevin dynamics in the overdamped regime with proper time acceleration.
Probabilistic solvers improve stability for stiff systems.
problem Performance penalties for small steps in stiff systems.
method Probabilistic exponential integrators that include fast linear dynamics in the prior.
result Proven L-stability and probabilistic error accounting.
We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
The paper analyzes a density estimator with exponential concentration.
problem Estimating integral functionals of continuous probability densities.
method Plug-in estimator for Hölder smooth densities on [0,1]d. result The estimator converges at rate $O \left( n^{-\fracβ{β+ d}}
ight)$ and is exponentially concentrated.
This paper summarizes closed-form relations for SE(3) maps and their derivatives.
problem Closed-form expressions for SE(3) maps and their derivatives are scattered in the literature.
method Summarizes and provides proofs for relevant closed-form relations of the exponential and Cayley map on SE(3).
result Provides an implicit generalized-alpha scheme for rigid/flexible multibody systems using the Cayley map.
Global stability proved for Navier-Stokes equations on hyperbolic space.
problem Stability of the Navier-Stokes equations on hyperbolic space.
method Proved global stability with exponential decay rate for small initial data.
result Exponential decay rate of $μλ_\Def^{(3)}$ for Navier-Stokes equations on hyperbolic space.
Extends likelihood ratio exponential families to analyze various optimization methods.
problem Analyzing optimization methods like rate-distortion and information bottleneck.
method Linking geometric mixture paths to exponential families and using hypothesis testing.
result Provides a common mathematical framework for understanding these methods.
The paper studies exponential functionals of processes with independent increments and their moments.
problem Analyzing the moments of exponential functionals of processes with independent increments.
method Deriving recurrent integral equations for Mellin transforms and applying them to calculate moments.
result Explicit formulas for the moments of It and I∞, and precise number of finite moments of I∞. This paper gives a new definition of the Contou-Carrere symbol in terms of an exponential of a Chen iterated integral and proves the corresponding reciprocity law.
Researchers create integral representations for two-layer ReLU networks with quantitative bounds.
problem Approximating functions with two-layer ReLU networks using explicit integral representations.
method Developed integral representations involving harmonic extension and projection, providing L2 bounds. result Functions can be approximated with L2 errors independent of dimension or degree, depending on coefficients and distribution. Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
Quantum algorithm speeds up pricing of financial derivatives.
problem Pricing autocallable options efficiently.
method Integration-based exponential amplitude loading technique.
result 50x reduction in circuit depth for payoff component.
Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.
problem Infinite volume of moduli spaces of hyperbolic surfaces with cusps.
method Introduce exponential volume form exp(-W)Vol(K,L) where W is a function of hyperbolic areas.
result Exponential volume forms make moduli spaces finite and relevant to open string theory.
RHMC accelerates sampling from log-concave distributions.
problem Sampling from log-concave probability distributions efficiently.
method RHMC uses simulated Hamiltonian dynamics with random integration times.
result RHMC converges exponentially fast in KL divergence for log-concave distributions.
We consider a vector field X on a closed manifold which admits a Lyapunov one form. We assume X has Morse type zeros, satisfies the Morse--Smale transversality condition and has non-degenerate closed trajectories only. For a closed one form η, considered as flat connection on the trivial line bundle, the differen…
Method extends option valuation for 2D Lévy models.
problem Valuation of European options under 2-asset infinite-activity Lévy models.
method Developed numerical method extending Wang et al. (2007) for 1D to 2D, using Fourier transform for integral term and semi-Lagrangian theta-method for temporal discretization.
result Favourable second-order convergence for Normal Tempered Stable dynamics.
Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.
problem Analyzing Bergman projections with Gevrey weights.
method Extending direct approach to semiclassical asymptotics to Gevrey weights using Fourier integral operators.
result Gevrey symbol amplitude of asymptotic Bergman projection with Gevrey weights and Gevrey-type growth rate.
Estimates spectral risk measures from i.i.d. samples.
problem Estimating spectral risk measures from limited data.
method Numerical integration method for SRM estimation.
result Estimate concentrates exponentially for bounded support distributions.
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …
Guyon-Lekeufack model accurately predicts market volatility.
problem Modeling and predicting market volatility accurately.
method Path-dependent volatility model with weighted past price returns and squared volatility.
result Wellposedness of the coupled system of stochastic differential equations for all parameter values.
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.
ELNN uses neural networks for improved option pricing.
problem Inconsistent pricing of over-the-counter products and unacceptable outcomes in ANN-based models.
method ELNN integrates ANNs with the exponential Levy model, addressing issues with existing models.
result ELNN outperforms Merton and Kou models in fitting performance and stability of estimates.
We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an "integrated" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack …
Unified framework for understanding TVO and improving model learning.
problem Improving the tightness and efficiency of variational inference bounds.
method Exponential family interpretation and equal spacing in moment parameters.
result Unified framework and improved gradient estimator for TVO.
New method uses Monte Carlo estimation to approximate dually flat information geometry.
problem Intractable integral-based Bregman generators for dually flat statistical manifolds.
method Monte Carlo estimation of Bregman generators for dually flat information geometries.
result Monte Carlo Information Geometries (MCIG) allow practical use of Bregman algorithms.
Kernel thinning compresses distributions more effectively than i.i.d. sampling or standard thinning.
problem Efficiently compressing distributions for better sampling and integration accuracy.
method Introduces kernel thinning, a procedure that compresses an n-point approximation of a distribution into a sqrt(n)-point approximation with comparable integration error.
result Kernel thinning achieves a maximum discrepancy in integration error of O_d(n^(-1/2) sqrt(log n)) in probability for compactly supported distributions and O_d(n^(-1/2) (log n)^(d+1/2) sqrt(log log n)) for sub-exponential distributions.
VISTA learns causal structures by integrating local subgraphs, improving accuracy and efficiency.
problem Efficiently learning causal structures from high-dimensional observational data.
method VISTA decomposes the global causal structure learning problem into local subgraphs based on Markov Blankets, integrating them via a weighted voting mechanism.
result VISTA achieves notable improvements in accuracy and efficiency over existing methods.
Derives conditions for no arbitrage in financial markets with stochastic or diffusion models.
problem Existence and absence of arbitrage in financial markets with stochastic or diffusion models.
method Integral tests, martingale and strict local martingale properties of stochastic exponentials, Markov switching models.
result Conditions for the existence of minimal martingale measure and its preservation under Markov switching.
The paper approximates financial derivatives using neural networks and iterated integrals.
problem Approximating p-integrable financial derivatives. method Using iterated Stratonovich integrals and neural networks.
result Approximate solutions to the Lp-hedging problem. We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatl…
This paper generalizes beta divergence beyond its classical form associated with power variance functions of Tweedie models. Generalized form is represented by a compact definite integral as a function of variance function of the exponential dispersion model. This compact integral form simplifies derivations of many pr…
We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to supe…
Consider the one-parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non-symmetrical hyperboles. These generalizations coincide to the one obtained in the context of non-extensive thermostatistics. We show that these functions are suitable to describe and un…
Exact discrete mechanics for nonholonomic systems defined.
problem Discrete mechanics for nonholonomic systems.
method Constructing an exponential map and deriving exact discrete nonholonomic integrators.
result Reproduces continuous nonholonomic flow as discrete flow on constraint submanifold.
Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.
problem Optimal transport in unbounded settings with heavy-tailed distributions.
method Hilbert's projective metric for integrable functions of bounded growth, kernel integral operators as contractions.
result Exponential convergence of Sinkhorn's algorithm for light-tailed marginal distributions.
The paper explores risk-minimization for exponential additive models, providing mathematical expressions and numerical examples.
problem Risk-minimization in incomplete markets for exponential additive models.
method Derive explicit mathematical expressions for local risk-minimization strategies in exponential additive models.
result Provide necessary conditions for deriving expressions and confirm integrability conditions for specific models.
New information geometry from exponentially concave functions.
problem Optimal transport and information geometry.
method Information geometry tools applied to exponentially concave functions.
result New information geometry on the unit simplex.
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
The paper proves the stability of a flow in Schwarzschild space.
problem Stability of area preserving mean curvature flow in asymptotic Schwarzschild space.
method Demonstrates existence and exponential convergence of the flow for all time.
result The flow converges to a round sphere or a constant mean curvature surface.
Unified diffusive bounds for non-linear parabolic equations.
problem Proving diffusive upper bounds for parabolic equations.
method Simple exponential deformation argument.
result Unified diffusive upper bounds for a wide class of non-linear parabolic equations.
Paper explores robust estimators for kernel exponential families using smoothed total variation distances.
problem Outliers can severely impact classical estimators in statistical inference.
method Proposes smoothed total variation (STV) distance as a class of IPMs for robust estimation of kernel exponential families.
result STV-based estimators are robust against distribution contamination for kernel exponential families.
Exponential Lasso improves Lasso's robustness to outliers and heavy-tailed noise.
problem Lasso's sensitivity to outliers and heavy-tailed noise in high-dimensional statistics.
method Integrates an exponential-type loss function into the Lasso framework.
result Achieves strong statistical convergence rates robust to heavy-tailed contamination.