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…
Formula derived for FUP exponent in quasi-Fuchsian groups.
problem Quantifying the fractal uncertainty principle in higher dimensions.
method Explicit formula derivation for FUP exponent, dependence on porosity parameter quantified.
result Explicit essential spectral gap for quasi-Fuchsian groups in higher dimensions.
Deep CNNs are known to exhibit the following peculiarity: on the one hand they generalize extremely well to a test set, while on the other hand they are extremely sensitive to so-called adversarial perturbations. The extreme sensitivity of high performance CNNs to adversarial examples casts serious doubt that these net…
This study shows how optimizer choice affects adversarial robustness in neural networks.
problem Understanding and improving adversarial robustness in neural networks.
method Revisiting known results linking robust classifiers and minimum norm solutions, combining them with recent optimizer bias findings.
result Achieving both perfect standard accuracy and robustness with certain optimizers under specific conditions.
The paper optimizes model selection and parameter estimation for multi-dimensional Gaussian Mixture Models.
problem Learning and distinguishing multi-dimensional Gaussian Mixture Models with reliable model order selection and efficient estimation.
method The paper establishes an information-theoretic lower bound and proposes a thresholding-based estimation algorithm with a time complexity of O(k^2 n). It also introduces a gradient-based minimization method with PCA for high-dimensional cases.
result The proposed method matches the established lower bound in sample complexity and achieves optimal parametric convergence rate.
Improved MMD test for two-sample testing with random Fourier features.
problem Quadratic-time complexity of MMD test for large-scale analysis.
method Approximated MMD test using random Fourier features, investigating time-power trade-off.
result Sub-quadratic time complexity with same minimax separation rates as MMD test.
SGD learns sparse parities near computational limits with discontinuous phase transitions.
problem Learning sparse parities in deep learning.
method Empirical and theoretical analysis of SGD on sparse parities.
result SGD makes progress on sparse parities via Fourier gap, not Langevin-like mechanism.
We propose the Wasserstein-Fourier (WF) distance to measure the (dis)similarity between time series by quantifying the displacement of their energy across frequencies. The WF distance operates by calculating the Wasserstein distance between the (normalised) power spectral densities (NPSD) of time series. Yet this ratio…
We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps η_i's of the RW, f(η), is symmetric and its Four…
Neural networks outperform NTK on compositional tasks, revealing a complexity gap.
problem Understanding the performance gap between neural networks and NTK on tasks with compositional structure.
method Characterized Fourier and architectural complexities, and analyzed the minimax rates of the architecture class.
result The NTK estimator is exponentially sub-optimal compared to the minimax floor when complexities decouple.
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.
We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-L…
ARBITER learns SPX-VIX term structures without arbitrage constraints.
problem Arbitrage-free modeling of SPX-VIX term structures.
method Risk-neutral neural operator mapping market states to operator outputs enforcing static arbitrage constraints.
result ARBITER outperforms other models in derivatives term structure evaluation metrics.
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 nk−klog2k+k queries for k non-zero Fourier coefficients. We explain how neural networks learn to solve modular addition tasks.
problem How two-layer neural networks learn to solve modular addition tasks.
method Formalized a diversification condition during training, proving it allows the network to approximate the correct logic for modular addition.
result Neural networks can robustly identify the correct sum through phase symmetry and frequency diversification.
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.
Establish a unified framework for negative results in Fourier analysis.
problem Fourier restriction, Lp-improving, and Fourier decay problems method Quantitative understanding of geometric properties of measures
result Explicit obstructions to measure satisfying Fourier restriction, Lp-improving, or Fourier decay estimates 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.
This paper extends depth separation results to piece-wise oscillatory functions.
problem Approximating functions with piece-wise oscillatory structure using neural networks.
method Extends existing results to piece-wise oscillatory functions using proof strategy from (Eldan and Shamir, 2016).
result Approximation by one-hidden-layer networks holds at a poly(d) rate for functions with constant domain radius and oscillation rate.
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.
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.
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.
New Fourier metrics equivalent to Wasserstein distances in image processing.
problem Equivalence of Fourier-based and Wasserstein metrics in imaging problems.
method Extensions of Fourier-based metrics to handle different centers of mass and discrete measures, showing equivalence to Wasserstein distances.
result New Fourier metrics are equivalent to Wasserstein distances with explicit constants, improving runtime in image processing.
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.
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.
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.
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 …
QAOA matches classical tensor power iteration in spiked tensor model recovery.
problem Statistical estimation in spiked tensor model with computational gap.
method Analysis of QAOA performance on spiked tensor model.
result QAOA weak recovery threshold matches tensor power iteration.
Structured CNN designed using the prior information of problems potentially improves efficiency over conventional CNNs in various tasks in solving PDEs and inverse problems in signal processing. This paper introduces BNet2, a simplified Butterfly-Net and inline with the conventional CNN. Moreover, a Fourier transform i…
A hybrid framework for American option pricing under time-varying rough volatility.
problem Pricing American options under time-varying rough volatility.
method Signature method combined with gradient-boosted ensemble for Hurst parameter estimation, regime switch, and Random Fourier Features for acceleration.
result The proposed hybrid framework improves performance over fixed-roughness baselines and reduces duality gaps in some regimes.
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.
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.
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.
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.
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.
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 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.
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.
The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.
problem Calibrating and pricing options in polynomial Ornstein-Uhlenbeck volatility models.
method Analyzes Fourier-Laplace transforms, connects to Riccati equations, and develops numerical schemes.
result Establishes existence and solution for Riccati equations and provides efficient numerical methods.
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.
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.
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.
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.
Improves sparse recovery with non-linear Fourier features.
problem Sparse recovery challenges with non-linear Fourier features.
method Characterizes sufficient data points for perfect recovery.
result Sufficient data points depend on kernel matrix.
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.
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.
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.
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.