We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
Sublinear Algorithms for Sparse Fourier Analysis. Finally, we would like to point out that in the series of papers @cite_29 @cite_37 @cite_40 @cite_26 an approach from @cite_3 has been considered. The algorithms are based on so called isolation and group testing from theoretical computer science. It has been proven that one can recover a sparse trigonometric polynomial @math , @math , from @math randomly chosen samples on @math if These results have been generalized to some extend for multivariate trigonometric polynomials in @cite_37 @cite_40 @cite_26 . Moreover, @cite_37 indicates that @math samples suffice if the random sampling set possesses additional structure.
@cite_37 @cite_26 @cite_29 @cite_3 @cite_40
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Abstract We present a sublinear randomized algorithm to compute a sparse Fourier transform for nonequispaced data of a special type. More precisely, we address the situation where a signal S is known to consist of N equispaced time samples, of which only L N samples are available. If the ratio p = L N is much smaller than 1, the available data typically look like nonequispaced samples, with little or no visible trace of the equispacing of the full set of N samples. We extend an approach for equispaced data that was presented in [J. Zou, A.C. Gilbert, M. Strauss, I. Daubechies, Theoretical and experimental analysis of a randomized algorithm for sparse Fourier transform analysis, J. Comput. Phys. 211 (2006) 572–595]; the extended algorithm reconstructs, from the incomplete data, a near-optimal B-term representation R with high probability 1 − δ , in time and space poly ( B , log ( N ) , log ( 1 δ ) , e −1 ) , such that ‖ S − R ‖ 2 2 ⩽ ( 1 + e ) ‖ S − R opt B ‖ 2 2 , where R opt B is the optimal B-term Fourier representation of signal S. The sublinear poly ( log N ) time is compared to the superlinear O ( L 1 + ( d − 1 ) β log L ) time requirement of the present best known inverse nonequispaced fast Fourier transform (INFFT) algorithms, in the sense of weighted norm with the number of dimensions d and smoothness parameter β. Numerical experiments support the advantage of our algorithm in speed over other methods for sparse signals: it already outperforms the INFFT for large but realistic size N and works well even in the situation of a large percentage of missing data and in the presence of large noise. We analyze a sublinear RA@?SFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term Sparse representation R for a given discrete signal S of length N, in time and space poly(B,log(N)), following the approach given in [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002]. Its time cost poly(log(N)) should be compared with the superlinear @W(NlogN) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RA@?SFA, as presented in the theoretical paper [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002], turns out to be very slow in practice. Our main result is a greatly improved and practical RA@?SFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RA@?SFA constructs, with probability at least 1-@d, a near-optimal B-term representation R in time poly(B)log(N)log(1 @d) @e^2log(M) such that @?S-R@?"2^2=<(1+@e)@?S-R"o"p"t@?"2^2. Furthermore, this RA@?SFA implementation already beats the FFTW for not unreasonably large N. We extend the algorithm to higher dimensional cases both theoretically and numerically. The crossover point lies at N 70,000 in one dimension, and at N 900 for data on a NxN grid in two dimensions for small B signals where there is noise. Recently, it has been observed that a sparse trigonometric polynomial, i.e., having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using basis pursuit (BP) or orthogonal matching pursuit (OMP). In this paper, it is shown that recovery by a BP variant is stable under perturbation of the samples values by noise. A similar partial result for OMP is provided. For BP, in addition, the stability result is extended to (nonsparse) trigonometric polynomials that can be well approximated by sparse ones. The theoretical findings are illustrated by numerical experiments. 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.