Approximation by exponential sums revisited

Beylkin, Gregory; Monzón, Lucas

doi:10.1016/j.acha.2009.08.011

Cited by 162 publications

(187 citation statements)

References 26 publications

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“…Note that the emphasis in [3,4] is placed on approximate compressed representation of functions by linear combinations with only few exponentials. See also the impressive results in [5,6]. Recently, the two last named authors of this paper have investigated the properties and the numerical behavior of APM in [26], where only real-valued exponential sums (1.1) were considered.…”

mentioning

confidence: 99%

Nonlinear Approximation by Sums of Exponentials and Translates

Peter¹,

Potts²,

Tasche³

2011

SIAM J. Sci. Comput.

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Abstract. In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1-periodic window function as defined in section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.

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confidence: 99%

Nonlinear Approximation by Sums of Exponentials and Translates

Peter¹,

Potts²,

Tasche³

2011

SIAM J. Sci. Comput.

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“…All SNR are expressed in dB as SNR X = 10 log 10 jjX o jj 2 jjX − X o jj 2 , [10] where X o is the original clean dataset and X is the noisy dataset. SNR gains were computed as the difference of SNR between before and after denoising, also expressed as SNR gain = SNRX − SNR X , [11] whereX is the cleaned dataset.…”

Section: Methodsmentioning

confidence: 99%

“…For such specific signals, one class of denoising methods is based on modeling a sum of a fixed number of exponentials as devised by Prony (8). This was recently revisited and improved by Beylkin and Monzon (9,10).…”

mentioning

confidence: 99%

Efficient denoising algorithms for large experimental datasets and their applications in Fourier transform ion cyclotron resonance mass spectrometry

Chiron¹,

Agthoven

Kieffer

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

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Modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed. In many cases, the analysis of spectroscopic data involves the denoising of raw data before any further processing. Current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length, preventing their use on very large datasets. Taking advantage of recent progress on random projection and probabilistic algorithms, we developed a simple and efficient method for the denoising of very large datasets. Based on the QR decomposition of a matrix randomly sampled from the data, this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising. This procedure, called urQRd (uncoiled random QR denoising), strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size. The efficiency of these numerical tools is demonstrated on experimental data from high-resolution broadband Fourier transform ion cyclotron resonance mass spectrometry, which has applications in proteomics and metabolomics. We show that robust denoising is achieved in 2D spectra whose interpretation is severely impaired by scintillation noise. These denoising procedures can be adapted to many other data analysis domains where the size and/or the processing time are crucial. big data | noise reduction | matrix rank reduction | FT-ICR MS

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“…In fact, if we know in advance that there is a short exponential sum that can approximate f , we can use the algorithms developed in [10,11] (for continuous case) and [9] (for the discrete case). However, those works do not provide an easy characterization of the class of functions.…”

Section: Implementing Fouriermentioning

confidence: 99%

Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems

Deshpande

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk-averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). Under the assumption that there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above), 1 we can obtain the following approximation results for several important classes of utility functions:1. If the utility function µ is continuous, upper-bounded by a constant and lim x→+∞ µ(x) = 0, we show that we can obtain a polynomial time approximation algorithm with an additive error for any constant > 0.2. If the utility function µ is a concave increasing function, we can obtain a polynomial time approximation scheme (PTAS).3. If the utility function µ is increasing and has a bounded derivative, we can obtain a polynomial time approximation scheme.Our results recover or generalize several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.

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Approximation by exponential sums revisited

Cited by 162 publications

References 26 publications

Nonlinear Approximation by Sums of Exponentials and Translates

Nonlinear Approximation by Sums of Exponentials and Translates

Efficient denoising algorithms for large experimental datasets and their applications in Fourier transform ion cyclotron resonance mass spectrometry

Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems

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