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.
Study proves Volume Conjecture for Reshetikhin-Turaev invariants.
problem Volume Conjecture for Reshetikhin-Turaev invariants.
method Hyperbolic cone metrics and discrete Fourier transforms.
result Proves Volume Conjecture for most figure-8 knot configurations.
Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).
problem Investigating the geometry of a specific group of Fourier-integral operators.
method Defined a right-invariant pseudo-Riemannian metric on the group using renormalized traces of pseudo-differential operators.
result Extended the Hilbert-Schmidt Riemannian metric to the group.
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.
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.
We parametrize the space 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…
We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical L2 and Sobolev inner products, as well as those induced by translation-invariant reproducing kernels, for whic…
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.
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.
Supervised learning based on a deep neural network recently has achieved substantial improvement on speech enhancement. Denoising networks learn mapping from noisy speech to clean one directly, or to a spectrum mask which is the ratio between clean and noisy spectra. In either case, the network is optimized by minimizi…
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…
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.
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. A new graph signature invariant to graph automorphisms.
problem Graph symmetry and feature generation.
method Power spectrum signature derived from squared graph Fourier transform.
result Power spectrum signature is stable under graph perturbations.
Constructs smooth integrable magnetic systems on a two-torus.
problem Creating smooth magnetic systems on a two-torus with specific properties.
method Uses Nash-Moser implicit function theorem to find zeros of an action functional.
result Characterizes Zoll magnetic systems and proves their existence.
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.
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 of η invariants on lens spaces detects distinctions invisible to ordinary η.
problem Detecting distinctions in η invariants on lens spaces. method Spin-Fourier residues and equivariant η invariants. result Second derivative of the residual η germ is nonzero for some lens spaces. 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.
Let X be a compact connected strongly pseudoconvex CR manifold of dimension 2n+1,n≥1 with a transversal CR S1-action on X. In this paper we introduce the Quillen metric on the determinant line of the Fourier components of the Kohn-Rossi cohomology on X with respect to the S1-action. We study the behav…
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 …
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…
Proves existence of Yamabe metrics on conical manifolds with conical points and links.
problem Existence of Yamabe metrics on singular manifolds with conical points and links.
method Derives a counterpart of Aubin's result, uses conical links and Fourier analysis, adds lower-order correction to standard bubbles.
result Derives asymptotic expansions on the Yamabe quotient for generic type metrics.
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…
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.
Spectral methods predict long-term signals from linear and nonlinear systems.
problem Forecasting temporal signals from linear and nonlinear systems with arbitrary sampling.
method Introduces a spectral algorithm for linear signals and extends it to nonlinear systems using Koopman theory.
result The spectral methods achieve high accuracy in forecasting and uncertainty quantification.
We study the monopole Floer homology of a SOLV rational homology sphere Y from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable SOLV metrics on Y, small regular perturbations of the Seiberg-Witten equations do not admit irreducible solutions; in part…
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.
One of the most fundamental problems in machine learning is to compare examples: Given a pair of objects we want to return a value which indicates degree of (dis)similarity. Similarity is often task specific, and pre-defined distances can perform poorly, leading to work in metric learning. However, being able to learn …
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.