非等间距数据的稀疏傅里叶变换次线性算法

高效非局部线性图像去噪：采用非等距快速傅里叶变换与无矩阵预处理的双层优化

Efficient nonlocal linear image denoising: Bilevel optimization with Nonequispaced Fast Fourier Transform and matrix-free preconditioning

本文提出基于非归一化扩展高斯 ANOVA 核的双层优化非局部图像去噪方法，结合 NFFT 无矩阵矩阵-向量乘积、基变换与对角预处理，实现低存储、快速求解大规模稠密线性系统及参数学习。

带限函数傅里叶变换重构的最优密度补偿因子

Optimal density compensation factors for the reconstruction of the Fourier transform of bandlimited functions

本文将三角多项式精确重构方法推广至带限函数，重点研究采样密度补偿所需对角权重矩阵的计算，为MRI等非等距采样场景提供最优密度补偿因子。

Spectral decomposition of discrepancy kernels on the Euclidean ball, the special orthogonal group, and the Grassmannian manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of $\mathbb{R}^d$. For restrictions to the Euclidean ball in odd dimensions, to the rotation group $SO(3)$, and to the Grassmannian manifold $\mathcal{G}_{2,4}$, we compute the kernels' Fourier coefficients and determine their asymptotics. The $L_2$-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For $SO(3)$, the nonequisp…

Improved Approximation Guarantees for Sublinear-Time Fourier Algorithms

In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, $f : [0, 2 pi] -> C, in O(k^2 \cdot D^4)$ time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexiti…

INFFTM: 在MATLAB中快速评估三维傅里叶级数及其在量子涡旋再连接中的应用

INFFTM: Fast evaluation of 3d Fourier series in MATLAB with an application to quantum vortex reconnections

本文提出在MATLAB中利用NFFT快速评估三维傅里叶级数的方法，用于量子涡旋再连接问题，展示其在计算效率上的优势。

连续可积函数的傅里叶变换的NFFT计算

Computation of the Fourier transform for a continuous integrable function via NFFT

提出用三角采样多项式逼近连续傅里叶变换，并通过非均匀快速傅里叶变换（NFFT）高效计算，推导了ℓ∞误差界，数值实验验证了理论收敛速度。

散乱数据三角多项式插值的稳定性结果

Stability Results for Scattered Data Interpolation by Trigonometric Polynomials

提出一种基于共轭梯度法和非均匀快速傅里叶变换的快速算法，在温和假设下实现环面上散乱数据最优插值，总复杂度为O(M log M)。

High-Dimensional Sparse Fourier Algorithms

In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In ``Adaptive Sublinear Time Fourier Algorithm" by D. Lawlor, Y. Wang and A. Christlieb (2013), an efficient algorithm with $Θ(k\log k)$ average-case runtime and $Θ(k)$ average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth $N$, where $k$ is the number of significant modes such that $k\ll N$. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in the paper mentioned …

On the Design of Deterministic Matrices for Fast Recovery of Fourier Compressible Functions

We present a general class of compressed sensing matrices which are then demonstrated to have associated sublinear-time sparse approximation algorithms. We then develop methods for constructing specialized matrices from this class which are sparse when multiplied with a discrete Fourier transform matrix. Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods.

Multiscale High-Dimensional Sparse Fourier Algorithms for Noisy Data

We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $Θ(ds \log s)$ average-case runtime and $Θ(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, …

Deterministic Sparse Sublinear FFT with Improved Numerical Stability

In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (2018) for fast reconstruction of $M$-sparse vectors ${\mathbf x}$ of length $N= 2^J$, where we assume that all components of the discrete Fourier transform $\hat{\mathbf x}= {\mathbf F}_{N} {\mathbf x}$ are available. The sparsity of ${\mathbf x}$ needs not to be known a priori, but is determined by the algorithm. If the sparsity $M$ is larger than $2^{J/2}$, then the algorithm turns into a usual FFT algorithm with runtime ${\mathcal O}(N \log N)$. For $M^{2} < N$, the runtime of the algorithm is ${\mathcal O}(…

Rapidly Computing Sparse Legendre Expansions via Sparse Fourier Transforms

In this paper we propose a general strategy for rapidly computing sparse Legendre expansions. The resulting methods yield a new class of fast algorithms capable of approximating a given function $f:[-1,1] \rightarrow \mathbb{R}$ with a near-optimal linear combination of $s$ Legendre polynomials of degree $\leq N$ in just $(s \log N)^{\mathcal{O}(1)}$-time. When $s \ll N$ these algorithms exhibit sublinear runtime complexities in $N$, as opposed to traditional $Ω(N \log N)$-time methods for computing all of the first $N$ Legendre coefficients of $f$. Theoretical as well as numerical results dem…

Toeplitz Low-Rank Approximation with Sublinear Query Complexity

We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix $T \in \mathbb{R}^{d \times d}$. In particular, for any integer rank $k \leq d$ and $ε,δ> 0$, our algorithm makes $\tilde{O} \left (k^2 \cdot \log(1/δ) \cdot \text{poly}(1/ε) \right )$ queries to the entries of $T$ and outputs a rank $\tilde{O} \left (k \cdot \log(1/δ)/ε\right )$ matrix $\tilde{T} \in \mathbb{R}^{d \times d}$ such that $\| T - \tilde{T}\|_F \leq (1+ε) \cdot \|T-T_k\|_F + δ\|T\|_F$. Here, $\|\cdot\|_F$ is the Frobenius norm and $T_k$ is the op…

托普利茨矩阵的次线性时间低秩近似

Sublinear Time Low-Rank Approximation of Toeplitz Matrices

本文提出一种次线性时间算法，通过稀疏傅里叶变换和离网恢复技术，实现对带噪半正定托普利茨矩阵的近似最优低秩分解，解决了两个SODA开放问题。

A deterministic sparse FFT algorithm for vectors with small support

In this paper we consider the special case where a discrete signal ${\bf x}$ of length N is known to vanish outside a support interval of length $m < N$. If the support length $m$ of ${\bf x}$ or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute ${\bf x}$ from its discrete Fourier transform. In case of exact Fourier measurements we require only ${\cal O}(m \log m)$ arithmetical operations. For noisy measurements, we propose a stable ${\cal O}(m \log N)$ algorithm.

A sample efficient sparse FFT for arbitrary frequency candidate sets in high dimensions

In this paper a sublinear time algorithm is presented for the reconstruction of functions that can be represented by just few out of a potentially large candidate set of Fourier basis functions in high spatial dimensions, a so-called high-dimensional sparse fast Fourier transform. In contrast to many other such algorithms, our method works for arbitrary candidate sets and does not make additional structural assumptions on the candidate set. Our transform significantly improves upon the other approaches available for such a general framework in terms of the scaling of the sample complexity. Our…

稀疏傅里叶变换分析的随机化算法的理论与实验分析

Theoretical and Experimental Analysis of a Randomized Algorithm for Sparse Fourier Transform Analysis

提出并实现了一种改进的实用化随机化稀疏傅里叶分析算法（RAlSFA），在多项式时间内以高概率构造近最优B项稀疏表示，并在一维和二维中优于FFTW。