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.
We show that every knot has a checkerbord diagram and that every knot is the closure of a rosette braid. We define Fourier knots of type (n_1, n_2, n_3) as knots which have parametrizations where each coordinate function x_i(t) is a finite Fourier series of length n_i, and conclude that every knot is a Fourier knot of …
Polynomial expansions improve option pricing accuracy.
problem Efficiently pricing and Greeks in stochastic volatility models.
method Analytic series representations for European and exotic options.
result Polynomial expansions match Fourier transform accuracy.
The paper proposes a novel method for optimizing bounded functions using Fourier series and Ricci flow.
problem Optimizing bounded functions using Fourier series and Ricci flow.
method Approximating the initial manifold using Fourier series and center/boundary sampling. Iteratively evolving the manifold using geodesic hyper-spheres and inverse Ricci flow.
result The method allows for the optimization of high curvature regions and achieves potential global optima.
The paper derives statistics of multi-factor functions from their Fourier transforms.
problem Deriving statistics of multi-factor functions from Fourier transforms.
method Developed an m-Coefficient/Index Annihilation Theorem to analyze the moments of a function from its Fourier transform.
result The mth moment of a function becomes a series of terms, each with precisely m Fourier coefficients, and the indices sum to zero.
Improved stable pricing for European options using Fourier-Cosine series.
problem Lack of robustness in the COS method for various cases.
method Stable pricing of call options based on Fourier cosine series expansion.
result Stability demonstrated through error analysis and numerical examples.
High-dimensional inference for sparse spectral precision matrices
problem Inference on the spectral precision matrix at a fixed frequency
method Full likelihood-based inference using neighboring discrete Fourier transforms
result Simultaneous control of regularization, finite-sample truncation, and smoothing biases
Combines Fourier methods and RNNs for efficient time series prediction.
problem Efficiently processing and predicting time series data with memory and computational constraints.
method Uses short-time Fourier transform and weight reductions through low pass filtering in a Spectral RNN.
result Predicts time series data from chaotic systems and real-world data.
New method for approximating periodic kernels on high-dimensional data.
problem Inefficient modelling of periodicity in higher-dimensional problems.
method Index Set Fourier Series Features
result Significantly less predictive error compared to alternative methods.
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.
We create a polynomial with knot-like nodal lines.
problem Constructing a polynomial with a specific knot as its nodal set.
method Engineering a braid from finite Fourier series, then using it as the nodal set of a complex polynomial.
result For sufficiently small parameter, the nodal lines form the three-twist knot.
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.
New method for European option pricing faster and more robust.
problem Pricing European options efficiently and accurately.
method Fourier cosine series expansions for models with known characteristic functions.
result More robust and faster than the original COS method.
New method for pricing barrier options in time-dependent λ-SABR model.
problem Pricing barrier options in the time-dependent λ-SABR model.
method Modified integral transform method and Fourier-Bessel series solution.
result Semi-analytical solution for barrier options in λ-SABR model.
Improved language models learn complex distributions using Fourier series.
problem Capturing continuous structure in discrete token distributions.
method Introducing a Fourier head layer to model continuous structures.
result Significant improvements in performance across various tasks.
WF distance measures time series similarity via spectral density displacement.
problem Measuring similarity between time series.
method Wasserstein-Fourier distance between normalised power spectral densities.
result WF establishes as a general-purpose metric for time series.
Study identifies and analyzes three types of errors in learning Fourier operators.
problem Statistical, discretization, and truncation errors in learning Fourier operators.
method Analysis of a Discrete Fourier Transform (DFT) based least squares estimator.
result Established upper and lower bounds on statistical, discretization, and truncation errors.
Efficiently approximates time series correlation using Fourier transform and neural networks.
problem Efficiently approximating correlation in time series data.
method Embeds time series into a low-dimensional Euclidean space using Fourier transform and neural networks, ensuring accurate correlation approximation from Euclidean distance.
result Our method reduces approximation loss by half and improves top-k correlation search precision from 5% to 20%. Let M be a closed orientable Riemannian surface. Consider an SO(3)-connection A and a Higgs field Φ:M→so(3). The pair (A,Φ) naturally induces a cocycle over the geodesic flow of M. We classify (up to gauge transformations) cohomologically trivial pairs (A,Φ) with finite Fourier series in terms of a suita…
New scalable GP approximation using Fourier series decomposition.
problem Scalability and accuracy in Gaussian process approximations.
method Harmonic kernel decomposition (HKD) to decompose kernels orthogonally.
result Significantly outperforms standard variational methods in scalability and accuracy.
New method improves option pricing for non-smooth functions.
problem Inefficiency of Fourier techniques with non-smooth probability density functions.
method Singular Fourier-Padé (SFP) method
result Restores global spectral convergence rate and fast error convergence.
Developed Taylor series for muscle-finger system analysis.
problem Understanding the complex relationship between muscle activity and finger movement.
method Used Dendrite Net to develop Taylor series and construct relation spectrum.
result Found muscle synergy and coupling in hand movement.
Novel IMEX scheme solves financial PDEs with mixed derivatives.
problem Numerical approximations for financial PDEs with mixed derivatives.
method Second order finite volume IMEX Runge-Kutta scheme.
result Achieves true second order convergence with non-regular initial conditions.
New model predicts time series quantiles for nonstationary data.
problem Nonparametric probabilistic forecasting of nonstationary univariate time series.
method Composite Quantile Fourier Neural Network (QFNN) for extrapolation-based nonlinear quantile regression.
result Effective in providing high quality and accurate probabilistic predictions.
New methods improve translation-equivariant neural processes for modeling unknown functions.
problem Modeling unknown latent functions from irregularly sampled measurements.
method Volterra series and set Fourier convolutions to address translation-equivariance and efficiency.
result Improved translation-equivariant neural processes with analytical transparency and linear scalability.
iCOS method estimates risk-neutral densities and option prices without model assumptions.
problem Estimating risk-neutral densities and option prices without model assumptions.
method Leverages Fourier-cosine technique using option-implied cosine series coefficients, without model assumptions.
result Effective in extracting information from option prices under various market conditions.
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.
Improved barrier option pricing in Heston model using COS-BEM method.
problem Efficient barrier option pricing in the Heston model.
method Combining Fourier-cosine series (COS) method with Boundary Element Method (BEM).
result Significant computational efficiency improvement and BEM attractiveness for practitioners.
Discrete approach proves a new version of Wirtinger's inequality.
problem Classical Wirtinger's inequality for piece-wise functions.
method Theorem of classification of isometries and Fourier series development.
result New elementary proof of Wirtinger's inequality.
Study on estimating volatility of volatility using Fourier methods and provides insights into volatility dynamics.
problem Estimating the volatility of volatility (vol-of-vol) accurately and efficiently.
method Used Fourier methodology to estimate integrated volatility of volatility, bias-corrected and without bias-correction, comparing their asymptotic properties and accuracy.
result The bias-corrected estimator reaches the optimal rate n1/4, while the uncorrected estimator has a slower rate and smaller asymptotic variance. New method estimates Fourier transforms from finite data without periodicity assumptions.
problem Estimating Fourier transforms from discrete data points without periodicity assumptions.
method Gaussian process regression with gradient ascent method to estimate covariance function.
result Sharp and precise estimation of spectral density in noise-free and noisy signals.
Bayesian time series forecasting improves by dynamically adapting to recent information.
problem Lack of forgetting mechanism in signature kernel for time series forecasting.
method Introducing a novel forgetting mechanism for signature features using Random Fourier Decayed Signature Features (RFDSF) with Gaussian processes (GPs).
result Demonstrates superior performance compared to other GP-based alternatives and state-of-the-art probabilistic time series forecasting algorithms.
New method embeds correlation networks to reveal underlying time series patterns.
problem Analyzing correlation networks derived from time series data.
method Spectral embedding of noisy correlation networks, leveraging Fourier basis elements.
result Spectral embedding recovers true vertex-level latent representations under suitable assumptions.
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.
Study shows overparameterization helps in generalizing from smooth interpolants.
problem Understanding generalization in overparameterized linear models.
method Analysis of random Fourier series model with weighted trigonometric interpolation.
result Weighted trigonometric interpolation leads to lower generalization error in overparameterized scenarios.
This paper analyzes GANs using Fourier modes to stabilize training.
problem Stability and convergence issues in GAN training.
method Decompose GAN objective function into Fourier series and study dynamics.
result Convergent orbits in GANs are small perturbations of periodic orbits, justifying slow training.
A new algorithm computes Fourier coefficients for a specified range efficiently.
problem Inefficiency in FFT due to fixed output size for all applications.
method Fast Partial Fourier Transform (PFT) that allows specifying the range of Fourier coefficients to compute.
result PFT achieves significant speedup over state-of-the-art FFT algorithms for small output sizes.
Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
This paper uses topology to analyze financial price scaling.
problem Analyzing scaling characteristics in financial data.
method Decomposes financial price series into components of total variation.
result Demonstrates the suitability of total variation for fractal analysis.
Accelerates signature kernel computation for sequences.
problem Severe computational bottleneck in computing signature kernel.
method Random Fourier features to accelerate signature kernel computation.
result Uniform approximation guarantees for unbiased estimator with linear computation time.
FNN approximates functions and solves PDEs with periodic BCs.
problem Approximating and solving periodic functions and PDEs.
method Fourier neural network architecture with activation and loss functions.
result FNN can solve PDEs with periodic BCs and is interpretable.
Quantum ELMs use a quantum reservoir to learn from data, with limits on expressivity and scalability.
problem Understanding the limits of quantum ELMs for machine learning tasks.
method Decomposed QELM predictions into Fourier series to analyze expressivity and scalability.
result Expressivity of QELMs is limited by the number of Fourier frequencies and observables, and scalability is hindered by hardware noise and entanglement.
New model learns relative importance of multiple seasonal patterns in time series data.
problem Complex seasonal patterns in business time series data.
method Mixed hierarchical seasonality (MHS) model using Stan.
result Significant improvements in prediction error and predictive density compared to existing models.
This paper improves GANs for 1D data using reparameterizations.
problem Improving GAN performance for 1D data.
method Identifies and uses reparameterizations of Taylor and Fourier series.
result Reparameterized critics outperform standard GANs.
We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique o…
This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the perimeter of an oval is π times its average width, and provide a bound for the ra…
FEDformer combines Transformer with seasonal-trend decomposition for efficient long-term forecasting.
problem Transformer's inefficiency and inability to capture global time series views.
method Combines seasonal-trend decomposition with Transformer, exploiting Fourier basis for frequency enhancement.
result Reduces prediction error by 14.8% and 22.6% for multivariate and univariate time series, respectively.
Novel Fourier-based estimator reveals stochastic leverage effect in high-frequency data.
problem Analyzing the stochastic leverage effect in high-frequency data.
method A novel Fourier-based estimator of the stochastic leverage effect is defined and proven consistent.
result The magnitude of the stochastic leverage effect is detectable at high-frequency.