Multivariate exponential analysis from the minimal number of samples

Abstract

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.

A Convex Approach to Superresolution and Regularization of Lines in Images

“…We admittedly have these frequencies appearing on the first row x [1, :], but they are clustered and badly separated on this one. Consequently, it is better to use our procedure for estimating the angles, which exploits the other rows for larger index m, in order to space the frequencies before applying the Prony method; this is reminiscent of the strategies of decimation developed in [29,11,2]. Moreover, our proposed method to estimate the offsets η k deals with the rows and columns jointly and automatically preserves the correspondence, without having to restore it a posteriori.…”

“…The originality of our method is that we reduce the minimization over an infinite dictionary of lines to a semidefinite programming problem, taking advantage of the line structure in both directions of the grid. Although there are works in the same vein to recover 2-D point sources, or equivalently to estimate the parameters of 2-D exponentials [67,64,29,36], applying similar principles to the estimation of lines is not straightforward and is new, to our knowledge.…”

Efficient Multidimensional Diracs Estimation With Linear Sample Complexity

“…However, the sample complexity with the proposed approach therein scales exponentially in terms of the signal dimension [20]. Other sampling schemes with linear sample complexity have been proposed in [21], [22]. Finally, recent work on spectral compressed sensing (e.g., [23]) is not directly related with the multi-dimensional Dirac reconstruction.…”

Prony-Type Polynomials and Their Common Zeros

“…For example, Kunis et al [1] relies on the kernel basis analysis of the multilevel Toeplitz matrix of moments of f. In Cuyt et al [2], the exponential analysis has been considered as a Padé approximation problem. In contrast to the previous method, the algorithms developed in Diederichs and Iske [3] and Cuyt and Wen-Shin [4] use sampling of f along several lines in the hyperplane to obtain the univariate analog of the problem, which can be solved by classical one-dimensional approaches. Nevertheless, the stability of numerical solutions in the case of noise corruption still has a lot of open questions, especially when the number of parameters increases.…”

“…The use of Cantor pairing functions allows us to express bivariate Prony-type polynomials in terms of determinants and to find their exact algebraic representation. With respect to the number of samples the method of Prony-type polynomials is situated between the methods proposed in Kunis et al [1] and Cuyt and Wen-Shin [4]. Although the method of Prony-type polynomials requires more samples than Cuyt and Wen-Shin [4], numerical computations show that the algorithm behaves more stable with regard to noisy data.…”