In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.
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48 results for “Kernel integral operator”
Estimates kernel eigenvalues for compositional dot-product kernels.
problem Improving estimates for kernel eigenvalues.
method Eigenvalue decay estimates of integral operators associated with dot-product kernels.
result Improved estimates for kernel volumes in reproducing kernel Hilbert spaces.
Extends orbital integral evaluation to center of enveloping algebra.
problem Evaluate semisimple orbital integrals for arbitrary elements in the center of the enveloping algebra.
method Explicit geometric evaluation of Casimir operator to arbitrary elements in the center of the enveloping algebra.
result Extension of orbital integral evaluation to center of enveloping algebra.
Researchers transform equations and define integral operators on a ball.
problem Transforming equations from half space to ball.
method Identify Poisson kernel, define extension operator, prove inequalities.
result Uniqueness of extremal functions in limit case.
IGPs represent GPs using integral operators, improving regression efficiency.
problem Efficiently estimating kernel hyper-parameters and reducing prediction variance in GPs.
method Developed IGPs based on integral operators and a low-dimensional subspace for dimension reduction.
result Significant improvements in computational complexity and prediction variance.
Paper tackles conditional expectation estimation using compactification operators.
problem Estimating conditional expectations from product of two random variables.
method Operator theoretic approach using kernel integral operators in reproducing kernel Hilbert space.
result Solutions allow numerical approximation and convergence of data-driven implementations.
Study on heat kernel asymptotics and path integrals on Riemannian manifolds.
problem Investigating the short-time expansion of heat kernel on compact Riemannian manifolds.
method Formally expressing the heat kernel as a path integral and using Laplace's method.
result The lowest order term of the heat kernel's short-time expansion is given by the Fredholm determinant of the Hessian of the energy functional.
Paper explores RKHS properties for derivative and integral operators.
problem Establishing sufficient conditions for reproducing property in RKHS.
method Establishing reproducing property for combinations of composition operators.
result Provides framework for regularized learning algorithms involving function values, gradients, or operators.
Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
Kernel method embeds noisy datasets, capturing shared structures.
problem Limited power in capturing nonlinear structures, noisiness, high-dimensionality, and interpretability issues.
method Kernel spectral joint embeddings using duo-landmark integral operators.
result Consistent recovery of low-dimensional noiseless signals and convergence to eigenfunctions of integral operators.
Paper provides unbiased spectral moment estimates from finite data.
problem Challenges in estimating spectral moments from limited data.
method Dynamic programming approach to estimate spectral moments of kernel integral operator.
result Demonstrates consistency with theoretical spectra and practical utility in neural networks.
Study examines boundedness of oscillating singular integrals on specific Lie groups.
problem Investigating boundedness of oscillating singular integrals on Lie groups of polynomial growth.
method Presented kernel criteria in terms of sub-Riemannian structure and Fourier analysis.
result Extended classical oscillating conditions for boundedness of oscillating convolution operators.
New proof shows eventual positivity of integral operator for Hermitian functions.
problem Positivity of Hermitian algebraic functions on complex manifolds.
method Elementary and geometric proof of eventual positivity of an integral operator.
result Another proof of Quillen's positivstellensatz for Hermitian functions.
Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde \X$ its Oshima compactification, and $(π,\mathrm{C}(\widetilde \X))$ the regular representation of on $\widetilde \X$. We study integral operators on $\widetilde \X$ of the form , where is a rapidly falling function on …
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
The Fredholm integral equation of the first kind improves solutions for ill-posed supervised learning problems with limited data.
problem Ill-posed supervised learning problems with insufficient data.
method Using the Fredholm integral equation of the first kind (FIFK) with semi-supervised assumptions and MSDF methods.
result Improved accuracy and stability in solutions for ill-posed problems.
Paper proposes adaptive parameter selection for KGD algorithms.
problem Improving parameter selection for kernel-based gradient descent.
method Integrates bias-variance analysis with splitting method, introduces empirical effective dimension.
result Adaptive parameter selection strategy achieves optimal generalization error bound.
Develops a framework for learning nonlinear operators using Mercer kernels.
problem Learning nonlinear operators between infinite-dimensional spaces.
method Stochastic approximation framework with Mercer operator-valued kernels.
result Establishes dimension-free polynomial convergence rates for nonlinear operator learning.
A new data-adaptive prior stabilizes kernel learning in operators.
problem Learning kernels in operators from data is ill-posed due to nonlocal dependence.
method Introduces a data-adaptive prior to stabilize the Bayesian posterior mean.
result The data-adaptive prior achieves a stable posterior with small noise limits.
New method interpolates high-dimensional scattered data using kernel theory.
problem Scattered data in high-dimensional spaces defy traditional distributional assumptions.
method Kernel interpolation framework based on integral operator theory.
result Spectra of kernel matrices predict performance of interpolation methods.
In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on …
MCNO learns PDE solution operators using Monte Carlo sampling.
problem Learning solution operators for PDEs efficiently and flexibly.
method Directly learns kernel function using Monte Carlo sampling of input-output pairs.
result Competitive accuracy with efficient computational cost on 1D PDE benchmarks.
Generalizes neural networks for infinite-dimensional mappings, including PDE solutions.
problem Learning mappings between infinite-dimensional spaces and finite-dimensional approximations.
method Graph kernel network architecture with message passing for kernel integration.
result Competitive performance compared to state-of-the-art solvers for PDEs.
We propose to learn a kernel-based message operator which takes as input all expectation propagation (EP) incoming messages to a factor node and produces an outgoing message. In ordinary EP, computing an outgoing message involves estimating a multivariate integral which may not have an analytic expression. Learning suc…
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating with its first singular values.
result The new MMD distance measures the difference of two distributions with respect to local moments, where depends on singular values decay rate.
Develops vector-valued RKBS for neural networks and operators.
problem Understanding function spaces of -valued neural networks and neural operators.
method Defines and constructs vector-valued RKBS (vv-RKBS) without restrictive assumptions.
result Establishes Representer Theorem for neural architectures.
Adaptive rule improves kernel-based gradient descent performance.
problem Improving convergence speed of kernel-based gradient descent algorithms.
method Empirical effective dimension for stopping rule, learning theory analysis, integral operator approach.
result Optimal learning rates and iteration bounds for KGD with adaptive stopping rule.
Kernel-spectral embedding learns low-dim. structures from noisy data.
problem Learning low-dimensional nonlinear structures from high-dimensional noisy data.
method Adaptive bandwidth spectral embedding using integral operators.
result Convergence to noiseless embeddings and eigenfunctions of integral operators.
Kernel Conjugate Gradient achieves fast convergence rates for regression.
problem Statistical rates of convergence for kernel-based regression.
method Kernel Conjugate Gradient algorithm with early stopping for regularization.
result Upper bounds for and Hilbert norms, matching minimax lower bounds.
Quantitative Sobolev extensions and the Neumann heat kernel for integral Ricci curvature conditionsmath.DG
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…
The report analyzes infinite-dimensional output space regression.
problem Learning theory in vector-valued RKHS regression.
method Integral operator technique with spectral theory for non-compact operators.
result Results with minimal assumptions using Chebyshev's inequality.
Short proof of heat kernel asymptotics and convolution approximation.
problem Short time asymptotics and heat kernel approximation for Laplace type operators.
method Short time asymptotic expansion and convolution approximation of heat kernels.
result Approximation of heat kernel using repeated convolutions.
Fast Kernel Approximations for Latent Force Models and Convolved Multiple-Output Gaussian processesstat.ML
This paper uses random Fourier features to simplify latent force models and convolved Gaussian processes.
problem Expensive covariance matrix calculation in latent force models due to double integrals.
method Approximates double integrals using random Fourier features to obtain simpler analytical expressions.
result Simplified analytical expressions for covariance functions, leading to faster computation.
Robust learning method combines kernel smoothing and robust optimization.
problem Certifying robustness against distribution shifts in machine learning models.
method Adapting integral operator using supremal convolution for robustness, leveraging optimal transport.
result The method provides theoretical guarantees for certified robustness and competitive performance.
New algorithm improves regression error bounds and accelerates performance for low noise.
problem Nonparametric least square regression in RKHS with optimal error bounds.
method Kernel Truncated Randomized Ridge Regression (KTRRR) with optimal generalization error bounds.
result Faster finite-time and asymptotic rates on low noise problems.
New method learns kernels in nonlocal operators robustly.
problem Learning kernels in nonlocal operators is ill-posed.
method Nonparametric regression with Tikhonov regularization.
result Robust estimator of kernel yields homogenized model.
Study on minimal energy problems for Riesz kernels on manifolds.
problem Minimal energy problems for strongly singular Riesz kernels on manifolds.
method Formulation of natural regularization using Hadamard's partie finie integral operator and analysis of measures with finite energy in Sobolev space.
result The minimal energy problem admits a unique solution and is related to discrete minimal energy problems.
Study shows precise Szegö kernel behavior for CR manifolds with group actions.
problem Analyzing the Szegö kernel for CR manifolds with group actions.
method Used Fourier integral operators and CR moment maps to describe kernel behavior.
result Obtained precise asymptotic expansion for Szegö kernel components.
The goal of this article is twofold: in a first part, we prove Gaussian estimates for the heat kernel of Schr{ö}dinger operators delta + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz transform with potential d(delta+V) --1/2…
Study spectral invariants of singularities using Schrödinger operators.
problem Understanding the Milnor number of quasi-homogeneous singularities.
method Local index theory of Schrödinger operators and heat kernel expansions.
result Define torsion type invariants to study singularities.
Geometric properties of Toeplitz kernels relate to circle function injectivity.
problem Injectivity of Toeplitz operators on the unit circle.
method Linking Toeplitz operators to geodesics in Grassmann manifolds.
result Existence of geodesics in Grassmann manifolds connects Toeplitz operator injectivity.
Improved estimation of higher order integrals using shrinkage techniques.
problem Estimating higher order Bochner integrals in non-parametric settings.
method Shrinkage of U-statistic towards a target element, considering kernel degeneracy.
result Consistent shrinkage estimators with fast rates of convergence, even for non-degenerate kernels.
New wavelets use Monte Carlo for efficient discretization.
problem Efficiently discretizing continuous wavelets on general domains.
method Defined continuous wavelets via spectral calculus and proposed a Monte Carlo discretization.
result Convergence of Monte Carlo wavelets under natural regularity assumptions.
Paper studies adversarial training in RKHS, revealing a trade-off between robustness and generalization.
problem Adversarial training's trade-off between robustness and generalization.
method RKHS framework, kernel integral operator, noise-debiased procedure.
result Proposes a two-stage noise-debiased procedure to improve generalization rate and achieve minimax polynomial rate.
Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic …
The paper provides gradient estimates for heat kernels on manifolds with negative Ricci curvature.
problem Estimating gradients of heat kernels on manifolds with negative Ricci curvature.
method Pointwise and gradient estimates, uniform boundedness results for the heat operator of the Hodge Laplacian.
result Uniform boundedness results and gradient estimates for heat kernels and Hodge Laplacian.
Kernel Trace Distance: Quantum Statistical Metric between Measures through RKHS Density Operatorsstat.ML
A new distance metric compares probability distributions using kernel covariance operators.
problem Comparing probability distributions in machine learning tasks.
method Introduces a novel distance metric based on Schatten norm of kernel covariance operators.
result The new distance metric is more discriminative and robust to hyperparameters.