arXiv subjects

Gerlind Plonka

Publications and source records attributed to Gerlind Plonka.

At least 19 recordsLinked to original sources

arXiv2026

Approximation by short exponential sums with geometric error decay based on Gauss quadrature

We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0, \infty)$ and for $f_2(x)= {\mathrm e}^{-x^2/2\sigma}$ with $\sigma>0$ on ${\mathbb R}$ with geometric error decay ${\rho}^{-2N}$ for user-defined $N \ge 2$ and $\rho >1$. The approximations are built over consecutive intervals $[b_j, \, b_{j+1}) \subset [0, \infty)$, $j \in {\mathbb N}_{0}$, with interval lengths that depend on $\rho$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j, \, b_{j+1})$ are easily computed from the initial parameters on $[b_0, \, b_{1})$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\log(x)$ on $[1,\infty)$ and of the error function $\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.

math.NA↗

arXiv2025

Image reconstruction from structured subsampled 2D Fourier data

In this paper we study the performance of image reconstruction methods from incomplete samples of the 2D discrete Fourier transform. Inspired by requirements in parallel MRI, we focus on a special sampling pattern with a small number of acquired rows of the Fourier transformed image. We show the importance of the low-pass set of acquired rows around zero in the Fourier space for image reconstruction. A suitable choice of the width $L$ of this index set depends on the image data and is crucial to achieve optimal reconstruction results. We prove that non-adaptive reconstruction approaches cannot lead to satisfying recovery results. We propose a new hybrid algorithm which connects the TV minimization technique based on primal-dual optimization with a recovery algorithm which exploits properties of the special sampling pattern for reconstruction. Our method shows very good performance for natural images as well as for cartoon-like images for a data reduction rate up to 8 in the complex setting and even 16 for real images.

math.NA↗

arXiv2025

Differential approximation of the Gaussian by short cosine sums with exponential error decay

In this paper, we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We generalise and extend the differential approximation method proposed in [4, 40] to approximate $\mathrm{e}^{-t^{2}/2σ}$ in the weighted space $L^{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2ρ})$ where $σ, \, ρ>0$. We prove that the optimal frequency parameters $λ_1, \ldots , λ_{N}$ for this method in the approximation problem $ \min\limits_{λ_{1},\ldots, λ_{N}, γ_{1}, \ldots, γ_{N}}\|\mathrm{e}^{-\cdot^{2}/2σ} - \sum_{j=1}^{N} γ_{j} \, {\mathrm e}^{λ_{j} \cdot}\|_{L^{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2ρ})}$, are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of ${\mathcal O}(N^{3})$ operations. We derive a direct algorithm to solve this approximation problem based on a matrix pencil method for a special structured matrix. The entries of this matrix are determined by hypergeometric functions. For the weighted $L^{2}$-norm, we prove that the approximation error decays exponentially with respect to the length $N$ of the sum. An exponentially decaying error in the (unweighted) $L^{2}$-norm is achieved using a truncated cosine sum. Our new convergence result for approximation of Gaussian functions by exponential sums of length $N$ shows that exponential error decay rates $e^{-cN}$ are not only achievable for complete monotone functions.

math.NA↗

arXiv2025

Phasebook: A Survey of Selected Open Problems in Phase Retrieval

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

cs.IT↗

arXiv2024

MOCCA: A Fast Algorithm for Parallel MRI Reconstruction Using Model Based Coil Calibration

We propose a new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI. Our approach is based on a parameter model for the coil sensitivities using bivariate trigonometric polynomials of small degree. The derived MOCCA algorithm has low computational complexity of $O(N_c N^2 \log N)$ for $N \times N$ images and $N_c$ coils and achieves very good performance for incomplete MRI data. We present a complete mathematical analysis of the proposed reconstruction method. Further, we show that MOCCA achieves similarly good reconstruction results as ESPIRiT with a considerably smaller numerical effort which is due to the employed parameter model. Our numerical examples show that MOCCA can outperform several other reconstruction methods.

math.NA↗

arXiv2022

Spline Representation and Redundancies of One-Dimensional ReLU Neural Network Models

We analyze the structure of a one-dimensional deep ReLU neural network (ReLU DNN) in comparison to the model of continuous piecewise linear (CPL) spline functions with arbitrary knots. In particular, we give a recursive algorithm to transfer the parameter set determining the ReLU DNN into the parameter set of a CPL spline function. Using this representation, we show that after removing the well-known parameter redundancies of the ReLU DNN, which are caused by the positive scaling property, all remaining parameters are independent. Moreover, we show that the ReLU DNN with one, two or three hidden layers can represent CPL spline functions with $K$ arbitrarily prescribed knots (breakpoints), where $K$ is the number of real parameters determining the normalized ReLU DNN (up to the output layer parameters). Our findings are useful to fix a priori conditions on the ReLU DNN to achieve an output with prescribed breakpoints and function values.

math.NA↗

arXiv2022

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 matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

math.NA↗

arXiv2022

From ESPRIT to ESPIRA: Estimation of Signal Parameters by Iterative Rational Approximation

We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices which are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the matrix pencil method is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the matrix pencil method but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the matrix pencil method for noisy data and for signal approximation by short exponential sums.

math.NA↗

arXiv2021

Frame Soft Shrinkage Operators are Proximity Operators

In this paper, we show that the commonly used frame soft shrinkage operator, that maps a given vector ${\mathbf x} \in {\mathbb R}^{N}$ onto the vector ${\mathbf T}^{\dagger} S_γ {\mathbf T} {\mathbf x}$, is already a proximity operator, which can therefore be directly used in corresponding splitting algorithms. In our setting, the frame transform matrix ${\mathbf T} \in {\mathbb R}^{L \times N}$ with $L \ge N$ has full rank $N$, ${\mathbf T}^{\dagger}$ denotes the Moore-Penrose inverse of ${\mathbf T}$, and $S_γ$ is the usual soft shrinkage operator with threshold parameter $γ>0$. Our result generalizes the known assertion that ${\mathbf T}^{*} S_γ {\mathbf T}$ is the proximity operator of $\| {\mathbf T} \cdot \|_{1}$ if ${\mathbf T}$ is an orthogonal (square) matrix. It is well-known that for rectangular frame matrices ${\mathbf T}$ with $L > N$, the proximity operator of $\| {\mathbf T} \cdot \|_{1}$ does not have a closed representation and needs to be computed iteratively. We show that the frame soft shrinkage operator {${\mathbf T}^{\dagger} S_γ {\mathbf T}$} is a proximity operator as well, thereby motivating its application as a replacement of the exact proximity operator of $\| {\mathbf T} \cdot \|_{1}$. We further give an explanation, why the usage of the frame soft shrinkage operator still provides good results in various applications. In particular, we provide some properties of the subdifferential of the convex functional $Φ$ which leads to the proximity operator ${\mathbf T}^{\dagger} S_γ {\mathbf T}$ and show that ${\mathbf T}^{\dagger} S_γ {\mathbf T}$ approximates $\textrm{prox}_{\|{\mathbf T} \cdot\|_{1}}$.

math.FA↗

arXiv2021

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 order of $y$. The recovery is based on the observation that for $λ_{j} \not\in \frac{\mathrm i}{P} {\mathbb Z}$ the terms of $y$ possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [12]. If a sufficiently large set of $L$ Fourier coefficients of $y$ is available (i.e., $L > 2N+2$), then our recovery method automatically detects the number $M$ of terms of $y$, the multiplicities $n_{j}$ for $j=1, \ldots , M$, as well as all parameters $λ_{j}$, $j=1, \ldots , M$ and $ γ_{j,m}$ $j=1, \ldots , M$, $m=0, \ldots , n_{j}$, determining $y$. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method.

math.NA↗

arXiv2021

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 ${\mathcal O}(M^2 \, \log N)$. The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $M>20$ because of numerical instabilities, the modified algorithm is still numerically stable for $M=200$.

math.NA↗

arXiv2021

Optimal Rank-1 Hankel Approximation of Matrices: Frobenius Norm, Spectral Norm, and Cadzow's Algorithm

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.

math.NA↗

arXiv2020

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 though all terms of $f$ may be non-$P$-periodic, our reconstruction method requires at most $2K+2$ Fourier coefficients $c_{n}(f)$ to recover all parameters of $f$. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $K+1$ steps. The algorithm can also detect the number $K$ of terms of $f$, if $K$ is a priori unknown and $L>2K+2$ Fourier coefficients are available. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction of sparse non-periodic signals, rational approximation, AAA algorithm, barycentric representation, Fourier coefficients

math.NA↗

arXiv2020

A Tree-based Dictionary Learning Framework

We propose a new outline for adaptive dictionary learning methods for sparse encoding based on a hierarchical clustering of the training data. Through recursive application of a clustering method, the data is organized into a binary partition tree representing a multiscale structure. The dictionary atoms are defined adaptively based on the data clusters in the partition tree. This approach can be interpreted as a generalization of a discrete Haar wavelet transform. Furthermore, any prior knowledge on the wanted structure of the dictionary elements can be simply incorporated. The computational complexity of our proposed algorithm depends on the employed clustering method and on the chosen similarity measure between data points. Thanks to the multiscale properties of the partition tree, our dictionary is structured: when using Orthogonal Matching Pursuit to reconstruct patches from a natural image, dictionary atoms corresponding to nodes being closer to the root node in the tree have a tendency to be used with greater coefficients.

cs.LG↗

arXiv2020

Parseval Proximal Neural Networks

The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let $\mathcal H$ and $\mathcal K$ be real Hilbert spaces, $b\in\mathcal K$ and $T\in\mathcal{B}(\mathcal H,\mathcal K)$ have closed range and Moore-Penrose inverse $T^\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox}\colon\mathcal K\to\mathcal K$ the operator $T^\dagger\,\text{Prox} (T\cdot +b)$ is a proximity operator on $\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox} = S_λ\colon\ell_2 \rightarrow\ell_2$ and any frame analysis operator $T\colon\mathcal H\to\ell_2$ that the frame shrinkage operator $T^\dagger\, S_λ\,T$ is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $\mathbb R^d$ equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. Hence, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.

math.NA↗

arXiv2020

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 regards the change of operators from $A$ to $φ(A)$, where $φ$ is an analytic function, while $A$ and $φ(A)$ possess the same set of eigenfunctions. The goal is now to choose $φ$ such that the powers of $φ(A)$ are much simpler to evaluate than the powers of $A$. The second extension concerns the choice of the sampling functionals. We show, how new sets of different sampling functionals $F_{k}$ can be applied with the goal to reduce the needed number of powers of the operator $A$ (resp. $φ(A)$) in the sampling scheme and to simplify the acquisition process for the recovery method.

math.NA↗

arXiv2020

A sparse Fast Fourier Algorithm for Real Nonnegative Vectors

In this paper we propose a new fast Fourier transform to recover a real nonnegative signal ${\bf x}$ from its discrete Fourier transform. If the signal ${\mathbf x}$ appears to have a short support, i.e., vanishes outside a support interval of length $m < N$, then the algorithm has an arithmetical complexity of only ${\cal O}(m \log m \log (N/m)) $ and requires ${\cal O}(m \log (N/m))$ Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector ${\bf x}$. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if ${\bf x}$ has (almost) full support. The numerical stability of the proposed algorithm ist shown by numerical examples.

math.NA↗

arXiv2020

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

In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be recovered from function samples using generalized shift operators. We derive an ESPRIT-like algorithm for the generalized recovery method and show, that this approach can be directly used to reconstruct classical exponential sums from non-equispaced data. Furthermore, we derive a modification of Prony's method for sparse approximation with exponential sums which leads to a non-linear least-squares problem.

math.NA↗