New chaos formula simplifies variance calculation for Gaussian nodal volumes.
problem Analyzing the variance of Gaussian nodal volumes on Riemannian manifolds.
method Explicit Wiener-Itô chaos decomposition, reducing complexity from 2+2n to 4 Hermite polynomials. result New exact formula for variance and bounds, valid for arbitrary manifolds.
We introduce Hermite fractional financial markets, where market uncertainties are described by multidimensional Hermite motions. Hermite markets include as particular cases financial markets driven by multivariate fractional Brownian motion and multivariate Rosenblatt motion. Conditions for no-arbitrage and market comp…
New method uses spherical harmonics to simplify learning single-index models.
problem Learning single-index models with unknown one-dimensional projections.
method Proposes using spherical harmonics instead of Hermite polynomials to capture rotational symmetry.
result Characterizes the complexity of learning single-index models under arbitrary spherically symmetric input distributions.
New Hermite approximations accelerate convergence with adaptive coordinate transformations.
problem Accelerating convergence of spectral approximations for Hermite expansions.
method Using normalizing flows for adaptive coordinate transformations and deriving error estimates.
result Error estimates for Hermite expansions under adaptive coordinate transformations.
This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy subordinators. We construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigro…
Classified spaces in low dimensions.
problem Irreducible homogeneous almost Hermite-Lorentz spaces in low dimensions.
method Classification through complex dimension 3.
result Classification of spaces in low dimensions.
Gradient flow solves multi-index regression for high-dimensional Gaussian data.
problem Learning multi-index functions from high-dimensional Gaussian data.
method Two-timescale algorithm with non-parametric link function learning.
result Global convergence of Grassmannian population gradient flow dynamics.
Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
Study examines tangential real hypersurfaces on Hermite-like manifolds.
problem Characterizing real hypersurfaces on Hermite-like manifolds.
method Introduced tangential real hypersurfaces and derived main identities.
result Discussed contact metric structures in K-contact and cosymplectic cases.
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
Polynomial time algorithm learns depth-2 neural networks with ReLU activations.
problem Learning depth-2 neural networks with non-zero bias terms and general ReLU activations.
method Robust tensor decomposition of Hermite expansions.
result Polynomial time and sample efficient learning of depth-2 networks with ReLU activations.
This paper studies symplectic critical surfaces in Hermite surfaces.
problem Generalizing results about Kähler angle to the general case.
method Focuses on symplectic critical surfaces in Hermite surfaces.
result Provides a definition of symplectic critical surfaces in Hermite surfaces.
We present a new framework for Hermite fractional financial markets, generalizing the fractional Brownian motion and fractional Rosenblatt markets. Considering pure and mixed Hermite markets, we introduce a strategy-specific arbitrage tax on the rate of transaction volume acceleration of the hedging portfolio as the pr…
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
Hermite polynomials improve private data generation by reducing feature count.
problem Infinite-dimensional features in kernel mean embedding are impractical for private data generation.
method Replace random features with Hermite polynomial features, leveraging their ordered nature.
result Hermite polynomial features yield a more accurate approximation of kernel mean embedding with fewer features.
Paper develops federated GLMM algorithms for analyzing hierarchical data.
problem Analyzing hierarchical data with non-independent observations in a federated setting.
method Developed two federated GLMM algorithms using Laplace and Gaussian Hermite approximations.
result Federated GLMM can handle hierarchical data and achieve comparable or superior performance.
The article completes the research of two-point G2 Hermite interpolation problem with spirals by inversion of conics. A simple algorithm is proposed to construct a family of 4th degree rational spirals, matching given G2 Hermite data. A possibility to reduce the degree to cubic is discussed.
The study shows how certain geometries can be mapped to simpler structures.
problem Understanding the structure of geometries with specific properties.
method Reframing known submersion results and applying them to new contexts.
result Holonomy decompositions and holomorphic submersions are derived for specific geometries.
We give a classification, up to finite cover, of flat compact complete Hermite-Lorentz manifolds up to complex dimension 4.
Wide neural networks learn features under μP, identifying weights and decomposing support.
problem Feature learning in wide neural networks under μP. method Proving mean-field limit, characterizing identifiability, sparse-dictionary decomposition, and feature-learning-error decomposition.
result The triple (w∗,Dorb∗,S∗) identifies the natural learning cell of the architecture-data pair (σ,ρ). We define naturally Hermite-Lorentz metrics on almost-complex manifolds as special case of pseudo-Riemannian metrics compatible with the almost complex structure. We study their isometry groups.
New Hermite series estimator for Spearman rank correlation in non-stationary data.
problem Estimating time-varying Spearman rank correlation efficiently.
method Hermite series based sequential estimator for both stationary and non-stationary settings.
result Competitive performance compared to existing algorithms in simulations and real data.
Study compares parametric and Hermite-based models for option pricing.
problem Empirical performance of option price estimators.
method Examines parametric and nonparametric models, focusing on variance-gamma and Heston models.
result Hermite-based models can outperform Heston model in pricing errors.
Sequential quantile estimation refers to incorporating observations into quantile estimates in an incremental fashion thus furnishing an online estimate of one or more quantiles at any given point in time. Sequential quantile estimation is also known as online quantile estimation. This area is relevant to the analysis …
This research solves Hermite interpolation on manifolds using retractions.
problem Interpolating data on non-Euclidean spaces with matching derivatives.
method Proposes a novel procedure using retractions for Hermite interpolation on various manifolds.
result Establishes the well-posedness of the method and extends Hermite interpolation results to manifolds.
Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h-convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
Existence of metrics on non-Kähler varieties, generalizing previous work.
problem Existence of metrics on non-Kähler varieties.
method Definition of slope stability and existence of singular Hermite-Einstein metrics.
result Existence and uniqueness of singular Hermite-Einstein metrics for slope-stable sheaves.
Study proposes a method to construct copulas using corrected Hermite polynomial expansion for estimating foreign exchange volatility.
problem Estimating cross foreign exchange volatility with complex correlation structures.
method Applying corrections to the finite sum of multivariate Hermite polynomial expansions to construct copulas.
result The proposed copula method accurately reproduces the volatility smile of cross currency pairs.
Extends classical stability results to new geometric settings.
problem Stability of holomorphic vector bundles on complex manifolds.
method Introduces (ω,Ω)-Hermite-Einstein and (ω,Ω)-stable conditions. result Generalised Hermite-Einstein condition implies (ω,Ω)-semi-stability. Two methods for interpolating manifold-valued data are presented.
problem Interpolating manifold-valued functions with derivative constraints.
method Two approaches: weighted Riemannian barycenters and tangent space interpolation.
result Both methods are valid and perform well with numerical examples.
New stability criteria for vector bundles linked to Hermite-Einstein geometry.
problem Stability of higher-rank vector bundles and their moduli spaces.
method Introducing m-positivity and a smooth function for coherent subbundles, linking to Hermite-Einstein geometry. result Hermite-Einstein bundles are uniformly semi-stable, and new stability conditions are established.
New method uses Hermite polynomials for American option valuation.
problem Valuation of American options with complex jump-diffusion dynamics.
method Hermite polynomial expansions of transition density and early exercise premium.
result Converging approximations to true option prices and exercise boundaries.
The study approximates option prices using Hermite polynomials without assuming a specific distribution.
problem Approximating option prices without assuming a specific distribution of returns.
method Approximating the logarithmic return's density by a linear combination of rescaled Hermite polynomials.
result Empirical results suggest reasonable performance for options with moderate strike prices.
Rectified Linear Units (ReLUs) are among the most widely used activation function in a broad variety of tasks in vision. Recent theoretical results suggest that despite their excellent practical performance, in various cases, a substitution with basis expansions (e.g., polynomials) can yield significant benefits from b…
Algorithm solves American options with regime-switching using multigrid and compact finite difference.
problem Pricing American put options with regime-switching.
method Multigrid iterative algorithm based on compact finite difference schemes and Hermite interpolation.
result The algorithm provides a fast and efficient tool for pricing American put options with regime-switching.
On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of…
The article describes canonical metrics on holomorphic fibre bundles.
problem Existence of canonical metrics on isotrivial Kähler fibrations.
method Induced from Hermite--Einstein connections on holomorphic principal bundles.
result Existence of optimal symplectic connections when principal bundles are polystable.
In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre p…
Derives a series expansion for Asian option pricing with polynomial jump-diffusion moments.
problem Pricing Asian options with polynomial jump-diffusion processes.
method Uses Hermite polynomials and moments of the underlying process for closed-form computation.
result Explicit computation of Greeks and accurate series expansion for Asian options.
Improved option pricing for SABR model using Gauss-Hermite quadrature.
problem Improving accuracy of option pricing in the SABR model.
method Using Gauss-Hermite quadrature for numerical integration of the integrated variance.
result New method provides accurate option prices across all strike prices.
A Lorentzian manifold is defined here as a smooth pseudo-Riemannian manifold with a metric tensor of signature ((2n +1, 1)). A Robinson manifold is a Lorentzian manifold (M) of dimension (\geqslant 4) with a subbundle (N) of the complexification of (TM) such that the fibers of (N\to M) are maximal totally null (isotrop…
NQE uses quantile regression for fast SBI with cubic Hermite splines.
problem Efficient Bayesian inference for complex models with limited data.
method Neural Quantile Estimation (NQE) learns quantiles autoregressively and interpolates them using cubic Hermite splines.
result NQE achieves state-of-the-art performance on various benchmark problems.
We develop Fourier methods to expand translation-invariant kernels.
problem Constructing orthonormal expansions for translation-invariant kernels.
method Fourier analytic technique to derive explicit expansions.
result Explicit expansions for various kernels (Matérn, Cauchy, Gaussian).
A method for classifying points with minimal queries using Hermite polynomials.
problem Classifying points from an unknown probability measure with minimal label queries.
method Convex combination of conditional probabilities, Hermite polynomial kernel for hierarchical support estimation.
result The method achieves high F-score for classification in hyper-spectral images and MNIST. We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and on…
Efficient method for pricing Bermudan moving average options using GPR-GHQ.
problem High-dimensional pricing of Bermudan moving average options in energy markets.
method Gaussian Process Regression and Gauss-Hermite quadrature.
result GPR-GHQ method efficiently handles long windows and high dimensionality.
We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle F over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on F coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…
Efficient method for high-dimensional American option pricing and hedging.
problem High-dimensional American option pricing and hedging.
method Gradient-enhanced sparse Hermite polynomial expansions combined with least squares Monte Carlo.
result Outperforms state-of-the-art methods in high dimensions with comparable computational cost.