We analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the roots of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony's approach to multivariate decomposition problems is presented , exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial-exponential decomposition , using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carath 'e odory-Fej 'e r decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process and to decompose polynomial-exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial-exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial-exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.
For the sparse reconstruction problem from a discrete set of frequencies, it is shown in @cite_1 that the @math minimization provides a solution, for enough Fourier coefficients (at least @math ) chosen at random. As shown in @cite_69 , this problem can also be solved by a Prony-like approach, using only @math Fourier coefficients.
@cite_69 @cite_1
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This paper improves upon best-known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r-sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size k(r, n) = O(r log(n)·log 2 (r) log(r logn)) = O(r log 4 n ). A random set Ω satisfies this with high probability. This estimate is optimal within the log log n and log 3 r factors. We also give a relatively short argument for a similar problem with k(r, n) ≈ r[12 + 8 log(n r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short. This paper proves best known guarantees for exact reconstruction of a sparse signal f from few non-adaptive universal linear measurements. We consider Fourier measurements (random sample of frequencies of f) and random Gaussian measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly non-convex problem to a convex problem, and then solve it as a linear program. What are best guarantees for the reconstruction problem to be equivalent to its convex relaxation is an open question. Recent work shows that the number of measurements k(r,n) needed to exactly reconstruct any r-sparse signal f of length n from its linear measurements with convex relaxation is usually O(r poly log (n)). However, known guarantees involve huge constants, in spite of very good performance of the algorithms in practice. In attempt to reconcile theory with practice, we prove the first guarantees for universal measurements (i.e. which work for all sparse functions) with reasonable constants. For Gaussian measurements, k(r,n) lsim 11.7 r [1.5 + log(n r)], which is optimal up to constants. For Fourier measurements, we prove the best known bound k(r, n) = O(r log(n) middot log2(r) log(r log n)), which is optimal within the log log n and log3 r factors. Our arguments are based on the technique of geometric functional analysis and probability in Banach spaces. We consider the problem of robustly recovering a @math -sparse coefficient vector from the Fourier series that it generates, restricted to the interval @math . The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of @math ) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm @math , we show upper and lower bounds on the minimax error rate that both scale like @math , providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity @math , or equivalently from comparing @math to the so-called @math -spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math .