arXiv subjects

Wen-shin Lee

Publications and source records attributed to Wen-shin Lee.

5 recordsLinked to original sources

arXiv2023

Multiscale matrix pencils for separable reconstruction problems

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the nonlinear parameters. The matrix pencil method, which reformulates the problem statement into a generalized eigenvalue problem for the nonlinear parameters and a structured linear system for the linear parameters, is generally considered as the more stable method to solve the problem computationally. In Section 2 the matrix pencil associated with the classical complex exponential fitting or sparse interpolation problem is summarized and the concepts of dilation and translation are introduced to obtain matrix pencils at different scales. Exponential analysis was earlier generalized to the use of several polynomial basis functions and some operator eigenfunctions. However, in most generalizations a computational scheme in terms of an eigenvalue problem is lacking. In the subsequent Sections 3--6 the matrix pencil formulation, including the dilation and translation paradigm, is generalized to more functions. Each of these periodic, polynomial or special function classes needs a tailored approach, where optimal use is made of the properties of the parameterized elementary or special function used in the sparse interpolation problem under consideration. With each generalization a structured linear matrix pencil is associated, immediately leading to a computational scheme for the nonlinear and linear parameters, respectively from a generalized eigenvalue problem and one or more structured linear systems. Finally, in Section 7 we illustrate the new methods.

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arXiv2020

VEXPA: Validated EXPonential Analysis through regular sub-sampling

We present a procedure that adds a number of desirable features to standard exponential analysis algorithms, among which output reliability, a divide-and-conquer approach, the automatic detection of the exponential model order, robustness against some outliers, and the possibility to parallelize the analysis. The key enabler for these features is the introduction of uniform sub-Nyquist sampling through decimation of the dense signal data. We actually make use of possible aliasing effects to recondition the problem statement rather than that we avoid aliasing. In Section 2 the standard exponential analysis is described, including a sensitivity analysis. In Section 3 the ingredients for the new approach are collected, of which good use is made in Section 4 where we essentially bring everything together in what we call VEXPA. Some numerical examples of the new procedure illustrate in Section 5 that the additional features are indeed realized and that VEXPA is a valuable add-on to any stand-alone exponential analysis. While returning a lot of additional output, it maintains a favourable comparison to the CRLB of the underlying method, for which we here choose a matrix pencil method. Moreover, the output reliability of VEXPA is similar to that of atomic norm minimization, whereas its computational complexity is far less.

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arXiv2018

How to get high resolution results from sparse and coarsely sampled data

Sampling a signal below the Shannon-Nyquist rate causes aliasing, meaning different frequencies to become indistinguishable. It is also well-known that recovering spectral information from a signal using a parametric method can be ill-posed or ill-conditioned and therefore should be done with caution. We present an exponential analysis method to retrieve high-resolution information from coarse-scale measurements, using uniform downsampling. We exploit rather than avoid aliasing. While we loose the unicity of the solution by the downsampling, it allows to recondition the problem statement and increase the resolution. Our technique can be combined with different existing implementations of multi-exponential analysis (matrix pencil, MUSIC, ESPRIT, APM, generalized overdetermined eigenvalue solver, simultaneous QR factorization,...) and so is very versatile. It seems to be especially useful in the presence of clusters of frequencies that are difficult to distinguish from one another.

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arXiv2017

Multivariate Exponential Analysis from the Minimal Number of Samples

The problem of multivariate exponential analysis or sparse interpolation has received a lot of attention, especially with respect to the number of samples required to solve it unambiguously. In this paper we show how to bring the number of samples down to the absolute minimum of $(d+1)n$ where $d$ is the dimension of the problem and $n$ is the number of exponential terms. To this end we present a fundamentally different approach for the multivariate problem statement. We combine a one-dimensional exponential analysis method such as ESPRIT, MUSIC, the matrix pencil or any Prony-like method, with some linear systems of equations because the multivariate exponents are inner products and thus linear expressions in the parameters.

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arXiv2017

A hybrid Fourier-Prony method

The FFT algorithm that implements the discrete Fourier transform is considered one of the top ten algorithms of the $20$th century. Its main strengths are the low computational cost of $\mathcal{O}(n \log n$) and its stability. It is one of the most commonly used algorithms to analyze signals with a dense frequency representation. In recent years there has been an increasing interest in sparse signal representations and a need for algorithms that exploit such structure. We propose a new technique that combines the properties of the discrete Fourier transform with the sparsity of the signal. This is achieved by integrating ideas of Prony's method into Fourier's method. The resulting technique has the same frequency resolution as the original FFT algorithm but uses fewer samples and can achieve a lower computational cost. Moreover, the proposed algorithm is well suited for a parallel implementation.

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