Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
New discrepancy function compares discrete probability measures considering space geometry.
problem Comparing discrete probability measures in a geometrically meaningful way.
method Proposes the Fourier Discrepancy Function, proving convexity, differentiability, and providing gradient formula.
result Proves the Fourier Discrepancy is convex, twice differentiable, and provides an explicit gradient formula.
Paper proves Fourier transform for valuations, simplifying previous work.
problem Existence of isomorphism for translation-invariant smooth valuations.
method Directly describes Alesker's isomorphism in terms of Fourier transform on functions.
result Simple proofs of Alesker's Fourier transform properties, including a previously conjectured result.
A new Fourier model improves ODE prediction.
problem Improving the accuracy of ODE solutions, especially for periodic functions.
method Constructing a Fourier state space model and a hybrid model combining Taylor and Fourier methods.
result The hybrid model can predict ODE solutions more accurately, especially for periodic functions.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
Improved electrical load forecasting model using Fourier-enhanced RNN.
problem Electrical load time series downscaling with high accuracy and low error.
method Combines recurrent neural network with Fourier seasonal embeddings and self-attention.
result Significantly reduces RMSE across different time horizons compared to existing methods.
The paper tackles drift identification in Lévy α-stable stochastic systems, proposing a Fourier space approach.
problem Estimating the drift field of a stochastic differential equation driven by Lévy α-stable noise.
method Fourier space approach, parameterizing the drift field using Fourier series, minimizing a loss function with gradients computed via the adjoint method.
result The method is capable of learning drift fields in qualitative and/or quantitative agreement with ground truth fields.
New method for optimizing risk in financial models using Fourier transforms.
problem Optimizing risk in financial models with multi-period mean-CVaR.
method Strictly monotone 2D integration scheme via Fourier-trained transition kernels.
result Established robust and accurate optimization method for financial models.
RNNs solve modular addition tasks using low rank and sparse Fourier structures.
problem Solving modular addition tasks with recurrent neural networks.
method Identified low rank structures and sparse Fourier representations in RNN weights.
result RNNs robust to removing individual frequencies but degrade with more ablation.
New Fourier-based diffusion model improves high-frequency generation quality.
problem Diffusion models struggle with high-frequency details.
method Analyzed and modified the forward process in Fourier space to equalize noise corruption across frequencies.
result Improved generation quality for high-frequency components.
The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.
problem Classifying 3-manifolds up to integer homology cobordism.
method Exploring the Fourier transform of Heegaard Floer d-invariants.
result Lens spaces are cancellable in the monoid of 3-manifolds up to integer homology cobordism.
New Fourier analysis method for non-uniform Boolean hypercube.
problem Non-uniform probability measures on the Boolean hypercube.
method ANOVA-based decomposition, explicit basis, least squares problem.
result Generalization of Fourier analysis for arbitrary probability measures.
Lecture notes on kernel functions and Random Fourier Features.
problem Understanding and approximating kernel functions in machine learning.
method Mathematical background and proofs of concentration results.
result Estimation of error in Random Fourier Features approximation.
In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.
NFM models time-series data directly in the Fourier domain, achieving state-of-the-art performance.
problem Traditional time-series analysis focuses on the time domain, limiting flexibility.
method NFM models time-series data in the Fourier domain, using frequency extrapolation and interpolation.
result NFM achieves state-of-the-art performance on various time-series tasks.
Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifica…
The paper derives and proves the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
problem Deriving the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
method Employing the perspective of the functional equation satisfied by the classical Fourier transform, the paper derives the Helgason Fourier transform map and proves its properties.
result The Fourier transform is explicitly given and proven to be a map from vector bundle-valued differential forms to another vector bundle-valued differential form on the product space.
X-ray transform on H-type groups solved, revealing function injectivity.
problem Injectivity in sub-Riemannian geometry.
method Fourier Slice Theorem adapted to H-type groups.
result Integrable functions on H-type groups are uniquely determined by their integrals over geodesics.
Quantum computers can enhance spectral methods in machine learning.
problem Spectral methods are fundamental but challenging for classical models.
method Utilizing quantum Fourier Transform for spectral manipulations.
result Quantum computing can offer more efficient spectral design.
Enhances DGPs with adaptive RKHS Fourier features for better non-stationary pattern modeling.
problem Capturing complex non-stationary patterns in non-linear dynamical systems.
method Integrates ODE-based RKHS Fourier features into DGPs using convolution operations for adaptive amplitude and phase modulation. Uses a doubly stochastic variational inference framework.
result Improved predictive performance across various regression tasks.
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
FNSDA adapts to new dynamics via Fourier space adaptation.
problem Generalizing to unseen dynamical systems with limited data.
method Automatic partitioning of known environments in Fourier modes and adaptation of specific modes for new environments.
result FNSDA achieves superior or competitive generalization performance with reduced parameter cost.
Random Fourier features improve tabular deep learning convergence.
problem Tabular deep learning convergence issues.
method Random Fourier projections as a pre-processing step, projecting inputs into a fixed feature space.
result Random Fourier pre-processing accelerates tabular deep learning convergence.
Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the celebrated Random Fourier Feature methodology to get an approximation of operator-…
Quantum Fourier Transform aids machine learning inference.
problem Generalizing from finite data samples to ground truth.
method Inspired by quantum algorithms, uses Quantum Fourier Transform to expose invariant subspace for data comparison.
result Proposes a concrete implementation for machine learning applications leveraging symmetries.
This paper analyzes SHAP values using Fourier expansions for model interpretability.
problem Understanding and interpreting SHAP values in complex models.
method Developed a spectral framework using Fourier expansions for SHAP values in various model regimes.
result SHAP values are Lipschitz continuous in the deterministic regime and converge to Gaussian process values in the probabilistic regime.
As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the …
Study approximates operator learning for PDEs using Fourier multipliers.
problem Approximating operator behavior for PDE simulations.
method Approximation of operator symbols in Fourier domain using semi-norms.
result Identifies conditions for achieving predefined approximation error.
HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
We parametrize the space Z \mathcal{Z} Z of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. F…
New space for valuations in non-Archimedean setting with duality properties.
problem Developing a non-Archimedean analogue of classical valuation spaces.
method Construction of a new space of valuations with similar structures to classical spaces.
result The new space satisfies Poincaré duality and hard Lefschetz theorem.
Quantization and reduction studied for CR manifolds with group actions.
problem Quantization and reduction for CR manifolds with group actions.
method Consider a compact torsion free CR manifold X X X with a G G G -equivariant rigid CR line bundle L L L . The high tensor powers of L L L are studied, and a weighted G G G -invariant Fourier-Szegő operator projects onto the space of G G G -invariant CR sections. result Quantization commutes with reduction for sufficiently high tensor powers of the line bundle.
This work improves Fourier pricing for multi-asset options using RQMC with domain transformation.
problem Efficiently pricing multi-asset options in high dimensions with Fourier methods.
method Randomized quasi-Monte Carlo (RQMC) with domain transformation to handle singularities.
result RQMC with domain transformation provides accurate and scalable Fourier pricing for multi-asset options.
New algorithm for estimating multivariate quantiles using stochastic optimal transport.
problem Estimating multivariate quantiles from data.
method Stochastic algorithm for entropic optimal transport in Banach spaces, using Fourier coefficients.
result Almost sure convergence of the stochastic algorithm in infinite-dimensional Banach spaces.
A new method optimizes Fourier pricing for multi-asset options using adaptive quadrature.
problem Efficiently pricing multi-asset options in Lévy models.
method Optimized damping parameters and hierarchical adaptive quadrature.
result Significant speed-up in computational time for up to six dimensions.
Random Fourier features classification achieves fast learning rates with fewer features.
problem Improving classification efficiency with fewer features.
method Utilizing Lipschitz continuous loss functions and regularity conditions, the study reduces the number of features required for classification.
result Random Fourier features classification can achieve O ( 1 / n ) O(1/\sqrt{n}) O ( 1/ n ) learning rate with only Ω ( n log n ) Ω(\sqrt{n} \log n) Ω ( n log n ) features. A new SINC method for fast and accurate option pricing.
problem Computing option prices efficiently and accurately.
method SINC approach based on Shannon Sampling Theorem.
result SINC provides the most accurate and fast pricing computation.
Localized signal representation on graph bundles using Fourier analysis.
problem Representing signals on graph bundles with twists.
method Partition of unity and product factorization over the base graph.
result Lifted bases for signal spaces of graph bundle components.
A new method integrates Fourier basis expansion and mapping for improved time series forecasting.
problem Inconsistent starting cycles and series length issues in Fourier-based methods.
method Fourier Basis Mapping (FBM) method that integrates time-frequency features through Fourier basis expansion and mapping.
result FBM addresses inconsistencies and preserves temporal characteristics, achieving SOTA performance.
The paper bounds Fourier integral operators on Hardy spaces with specific conditions.
problem Bounding Fourier integral operators on Hardy spaces with given conditions.
method Using Hörmander classes and phase conditions, the paper establishes boundedness of Fourier integral operators.
result The Fourier integral operator is bounded from local Hardy space h p h^p h p to L p L^p L p under specified conditions. Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.
problem Analyzing light ray transform in pseudo-Euclidean space.
method Investigate normal operator, derive inversion formula, analyze as Fourier Integral Operator.
result Derive an inversion formula and prove stability estimates.
Study of η η η invariants on lens spaces detects distinctions invisible to ordinary η η η .
problem Detecting distinctions in η η η invariants on lens spaces. method Spin-Fourier residues and equivariant η η η invariants. result Second derivative of the residual η η η germ is nonzero for some lens spaces. Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp L p L^p L p bounds for eigenfunctions on products of rank-one symmetric spaces. Extend classical theory of affine processes to path-dependent setting
problem Path-dependent affine processes
method Introduce path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform
result Define path-dependent affine processes through their exponential-affine Fourier--Laplace transform and establish a characterization theorem
Fourier representation improves KSD for infinite-dimensional data.
problem Applying KSD to infinite-dimensional data.
method Combining measure equations with kernel methods for a Fourier representation of KSD.
result KSD can separate measures in infinite-dimensional Hilbert spaces.
New algorithms learn sparse set functions in non-orthogonal Fourier bases.
problem Learning sparse set functions in non-orthogonal Fourier bases.
method Novel algorithms using non-orthogonal Fourier transforms.
result At most n k − k log 2 k + k nk - k \log_2 k + k nk − k log 2 k + k queries for k k k non-zero Fourier coefficients. DAFNO learns surrogates for complex systems on irregular geometries.
problem Learning accurate surrogates for complex physical systems on irregular geometries.
method DAFNO incorporates a smoothed characteristic function in the integral layer architecture of FNOs, leveraging FFT for rapid computations.
result DAFNO achieves state-of-the-art accuracy on material modeling and airfoil simulation datasets.
Due to the isotropy d d d -dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the R R R -radius hyperboloid model of d d d -dimensional hyperbolic geometry with R > 0 R>0 R > 0 and d ≥ 2 d\ge 2 d ≥ 2 , we compute azimuthal Fourier expansions for a fundamental so…