Modifications of Prony's Method for the Recovery and Sparse Approximation of Generalized Exponential Sums

The Generalized Operator Based Prony Method

The generalized Prony method introduced by Peter & Plonka (2013) is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator $A$. However, this procedure requires the evaluation of higher powers of the linear operator $A$ that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that simplify the acquisition of the needed samples essentially and at the same time can improve the numerical stability of the method. The first extension reg…

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

In this paper we introduce two new algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz+Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matric…

稀疏逼近指数和的AAK理论应用

Application of the AAK theory for sparse approximation of exponential sums

本文提出利用AAK理论对离散信号进行最优ℓ¹和ℓ²逼近，通过Prony方法恢复指数和，再用AAK理论计算更短的指数和近似原信号。

Exact Reconstruction of Sparse Non-Harmonic Signals from Fourier Coefficients

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $f(t) = \sum_{j=1}^{K} γ_{j} \, \cos(2πa_{j} t + b_{j})$, where the frequency parameters $a_{j} \in {\mathbb R}$ (or $a_{j} \in {\mathrm i} {\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \cite{NST18}. For signal reconstruction we use a set of classical Fourier coefficients of $f$ with regard to a fixed interval $(0, P)$ with $P>0$. Even …

Exact Reconstruction of Extended Exponential Sums using Rational Approximation of their Fourier Coefficients

In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left( \sum_{m=0}^{n_j} \, γ_{j,m} \, t^{m} \right) {\mathrm e}^{2πλ_j t}$, where the frequency parameters $λ_{j} \in {\mathbb C}$ are pairwise distinct. For the reconstruction we employ a finite set of classical Fourier coefficients of $y$ with regard to a finite interval $[0,P] \subset {\mathbb R}$ with $P>0$. Our method requires at most $2N+2$ Fourier coefficients $c_{k}(y)$ to recover all parameters of $y$, where $N:=\sum_{j=1}^{M} (1+n_{j})$ denotes the o…

通过迭代有理逼近进行多元指数和的参数估计

Parameter estimation for multivariate exponential sums via iterative rational approximation

本文将单变量指数和的有理逼近重建算法扩展至多元情形，通过将多元有理插值问题降维为单变量问题并利用 AAA 方法求解，提出了基于稀疏网格和完整网格的两种新方法。

超分辨率恢复与实指数和分离的统一方法

A unified method for super-resolution recovery and real exponential-sum separation

本文提出一种统一方法，解决多变量超分辨率问题和盲源分离实值指数和问题，适用于荧光显微镜、天文观测及磁共振成像等应用。

紧集上算子学习的一致逼近

Uniform Approximation for Operator Learning on Compact Sets

该研究针对巴拿赫空间紧子集上的算子学习，刻画了其一致逼近性质，推导了恢复维度、网络参数数量及误差收敛速率的相关界，为算子学习的理论分析提供了关键支撑。

稀疏尖峰反卷积的精确支持恢复

Exact Support Recovery for Sparse Spikes Deconvolution

本文研究了稀疏尖峰反卷积在测度空间中的性质，证明在信噪比足够大时，总变分正则化能精确恢复尖峰位置，并展示其随噪声减小的收敛性。