The paper introduces new estimators for multivariate functions using Fourier methods.
problem Estimating multivariate functions like densities and regression functions.
method Monte Carlo estimators based on the Fourier integral theorem.
result Established rates of convergence for new estimators, often superior to existing methods.
We extend the Fourier cosine method to discrete probability distributions, achieving faster convergence rates.
problem Extending Fourier cosine method to discrete probability distributions.
method Spectral filters and convergence rates analysis.
result Spectral filters achieve one order faster convergence rates than previously recognized.
This work proves convergence of adaptive resampling for random Fourier features.
problem Sampling Fourier frequencies well for high-dimensional data.
method Data adaptive resampling of Fourier frequencies, asymptotically optimal.
result Proves convergence of adaptive resampling method for regression and classification problems.
Fourier methods fail to accurately approximate option Greeks in realistic market conditions.
problem Failure of Fourier pricing techniques to approximate Greeks in realistic market parameters.
method Used Fourier techniques like Carr-Madan formula, COS method, and Lewis formula to approximate Greeks, which failed in some market conditions.
result Empirically showed that Fourier methods completely fail to approximate Greeks in realistic market environments.
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.
Enhances Fourier estimator performance for asynchronous event-data.
problem Improving correlation and covariance estimation on event-data.
method Implement and test NUFFT methods with different averaging kernels.
result Demonstrates improved performance and relationship between averaging scales.
Improved Gaussian Process regression using TQFF over RFF and Gaussian QFF.
problem Limited performance of Quadrature Fourier Features (QFF) in approximating highly oscillatory functions.
method Developed Trigonometric Quadrature Fourier Features (TQFF) using a novel non-Gaussian quadrature rule.
result TQFF provides better approximation accuracy and fewer features compared to RFF and Gaussian QFF.
New method uses tensor decompositions to overcome the curse of dimensionality for large-scale learning.
problem Large-scale machine learning problems with kernel methods.
method Deterministic Fourier features combined with low-rank tensor decomposition for tensor product structure.
result Demonstrated consistent performance and superior results compared to random Fourier features.
New risk measure uses Fourier analysis of stock prices.
problem Identifying speculative behavior in financial products.
method Fourier analysis applied to stock price changes.
result Speculative behavior indicated by disproportionate price changes within one week.
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…
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.
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 method uses machine learning to optimize Fourier pricing methods.
problem Difficulty in tuning parameters for Fourier pricing methods.
method Learning tuning parameters of Fourier methods using machine learning.
result Very fast algorithms with full error control.
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
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.
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.
Stochastic control problems in finance often involve complex controls at discrete times. As a result numerically solving such problems, for example using methods based on partial differential or integro-differential equations, inevitably give rise to low order accuracy, usually at most second order. In many cases one c…
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.
A new NUFFT method speeds up option pricing for various strikes.
problem Efficiently pricing many options of the same maturity but different strikes.
method Non-uniform fast Fourier transform (NUFFT) applied to the COS method.
result Significantly faster computation of option prices.
DeepPhaseCut uses neural networks to improve Fourier phase retrieval.
problem Fourier phase retrieval from magnitude data.
method Unsupervised feed-forward neural network with cycleGAN training.
result Outperforms existing methods in Fourier phase retrieval.
Random Fourier features model reconstructs wind fields from sparse measurements.
problem Reconstructing wind fields from limited data.
method Random Fourier features approximating velocity field with adaptive sampling.
result Random Fourier features model outperforms benchmarks.
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.
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
Efficiently scales continuous kernels with sparse Fourier domain learning.
problem High computational and memory demands, spectral bias in continuous kernels.
method Sparse learning in the Fourier domain.
result Efficient scaling of continuous kernels, reduced computational and memory requirements, mitigated spectral bias.
Two ANOVA-based algorithms boost random Fourier feature models for function approximation.
problem Approximating high-dimensional functions with low-order interactions.
method Utilizes ANOVA decomposition to learn low-order functions and index sets of important variables.
result Significantly reduces approximation error compared to existing methods.
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.
A new density model using Fourier basis achieves better approximations and compression.
problem Approximating multi-modal 1D densities.
method Constrained Fourier basis model for end-to-end training.
result Lower cross entropy compared to deep factorized models.
Fourier methods have a long and proven track record as an excellent tool in data processing. As memory and computational constraints gain importance in embedded and mobile applications, we propose to combine Fourier methods and recurrent neural network architectures. The short-time Fourier transform allows us to effici…
We apply a new numerical method, the singular Fourier-Padé (SFP) method invented by Driscoll and Fornberg (2001, 2011), to price European-type options in Lévy and affine processes. The motivation behind this application is to reduce the inefficiency of current Fourier techniques when they are used to approximate piecew…
Periodicity is often studied in timeseries modelling with autoregressive methods but is less popular in the kernel literature, particularly for higher dimensional problems such as in textures, crystallography, and quantum mechanics. Large datasets often make modelling periodicity untenable for otherwise powerful non-pa…
A new method for nonstationary Gaussian processes using Fourier features.
problem Efficient simulation of nonstationary Gaussian processes with high-dimensional distributions.
method Discretizes the spectral representation of nonstationary processes, avoiding probability measure assumptions.
result An efficient low-rank approximation of nonstationary spectral densities, consistent and positive semi-definite.
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.
New quantization methods improve accuracy of Random Fourier Features.
problem Improving accuracy of Random Fourier Features for machine learning.
method Sigma-Delta and distributed noise-shaping quantization methods for 1-bit and low bit-depth quantization.
result Quantized RFFs allow high accuracy approximation of underlying kernels with polynomial error decay.
In this paper we present a slight modification of the Fourier estimation method of the spot volatility (matrix) process of a continuous Itô semimartingale where the estimators are always non-negative definite. Since the estimators are factorized, computational cost will be saved a lot.
Integrates Fourier features for faster Gaussian process regression.
problem Efficiently scaling Gaussian process regression to large datasets.
method Integrated Fourier features for Gaussian processes.
result Improves Gaussian process regression speed to O ( M 3 ) O(M^3) O ( M 3 ) for a broad class of kernels. 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 method slices and sums radial kernels faster.
problem Fast computation of large kernel sums in kernel methods.
method Random projections to 1D subspaces and QMC for selecting projections.
result QMC-slicing outperforms existing methods on test datasets.
Scalable kernel methods for large datasets using Fourier representations and NUFFT.
problem Cubic complexity in kernel methods limits their use on large-scale datasets.
method Fourier representation of kernels combined with NUFFT for O(n log n) complexity.
result Achieves minimax convergence rates and processes up to tens of billions of samples.
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.
New Fourier features improve high-precision approximation in large-scale problems.
problem Designing scalable, high-precision Fourier features for large-scale kernel methods.
method Introducing a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions.
result Improved approximation bounds with new Fourier features.
FourNet approximates financial transition densities using Fourier transforms.
problem Approximating transition densities in finance with high accuracy.
method FourNet is a novel FFNN with Gaussian activation, learning from characteristic functions.
result FourNet can approximate transition densities arbitrarily well with finite neurons.
The paper introduces a new volatility model using Fourier techniques for pricing and hedging.
problem Pricing and hedging of financial derivatives with stochastic volatility.
method A Fourier-based approach to price and hedge European and path-dependent options in a stochastic volatility model.
result The model includes and extends popular volatility models like Stein-Stein, Bergomi, and Heston.
The square root of Fredholm determinants causes numerical instabilities in option pricing models.
problem Numerical instabilities in Fourier-based option pricing for the Volterra Stein-Stein model.
method Characterization of determinant crossing behavior, derivation of transform to handle crossings, efficient algorithms.
result Significant improvement in accuracy and reduction in computational cost for Fourier-based pricing.
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.
Quantum methods improve option pricing accuracy.
problem Pricing financial derivatives using Monte Carlo integration.
method Hybrid classical-quantum methods using Fourier series and QML.
result Quantum methods achieve remarkable accuracy in option pricing.
We compare the CPU effort and pricing biases of seven Fourier-based implementations. Our analyses show that truncation and discretization errors significantly increase as we move away from the Black-Scholes-Merton framework. We rank the speed and accuracy of the competing choices, showing which methods require smaller …
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-…
Computing accurate estimates of the Fourier transform of analog signals from discrete data points is important in many fields of science and engineering. The conventional approach of performing the discrete Fourier transform of the data implicitly assumes periodicity and bandlimitedness of the signal. In this paper, we…