We consider the problem of computing the Walsh-Hadamard Transform (WHT) of some @math -length input vector in the presence of noise, where the @math -point Walsh spectrum is @math -sparse with @math scaling sub-linearly in the input dimension @math for some @math . Over the past decade, there has been a resurgence in research related to the computation of Discrete Fourier Transform (DFT) for some length- @math input signal that has a @math -sparse Fourier spectrum. In particular, through a sparse-graph code design, our earlier work on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm computes the @math -sparse DFT in time @math by taking @math noiseless samples. Inspired by the coding-theoretic design framework, proposed the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly computes the @math -sparse WHT in the absence of noise using @math samples in time @math . However, the SparseFHT algorithm explicitly exploits the noiseless nature of the problem, and is not equipped to deal with scenarios where the observations are corrupted by noise. Therefore, a question of critical interest is whether this coding-theoretic framework can be made robust to noise. Further, if the answer is yes, what is the extra price that needs to be paid for being robust to noise? In this paper, we show, quite interestingly, that there is no extra price that needs to be paid for being robust to noise other than a constant factor. In other words, we can maintain the same sample complexity @math and the computational complexity @math as those of the noiseless case, using our SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.
Among these works, the most recent progress in this direction is the sFFT (Sparse FFT) algorithm developed in the series of papers @cite_2 @cite_0 @cite_19 . Most of these algorithms are based on first isolating (i.e., hashing) the non-zero DFT coefficients into different bins, using specific filters or windows that have good' (concentrated) support in both time and frequency. The non-zero DFT coefficients are then recovered iteratively, one at a time. The filters or windows used for the binning operation are typically of length @math . As a result, the sample complexity is typically @math or more, with potentially large big-Oh constants as demonstrated in @cite_1 . Then, @cite_19 further improved the @math -D DFT algorithm for the special case of @math , which reduces the sample complexity to @math and the computational complexity to @math , albeit with a constant failure probability that does not vanish as the signal dimension @math grows. On this front, the deterministic algorithm in @cite_26 is shown to guarantee zero errors but with complexities of @math . More recently, @cite_4 develops a deterministic algorithm for computing a sparse WHT in time @math with an arbitrary constant @math .
@cite_26 @cite_4 @cite_1 @cite_0 @cite_19 @cite_2
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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. 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. Abstract In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; we discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold M . The quasi-affine system is formed by generalized dilations and shifts of a finite collection of wavelet functions Ψ : = ψ j : 1 ≤ j ≤ r ⊂ L 2 ( R ) . We further require that ψ j is generated by some refinable function ϕ with mask a j . We present the condition needed for the masks a j : 0 ≤ j ≤ r , as well as regularity conditions needed for ϕ and ψ j , so that the associated quasi-affine system generated by Ψ is a tight frame for L 2 ( M ) . The condition needed for the masks is a simple set of algebraic equations which are not only easy to verify for a given set of masks a j , but also make the construction of a j entirely painless. Then, we discuss how the transition from the continuum (manifolds) to the discrete setting (graphs) can be naturally done. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately by proposing the fast tight wavelet frame transforms for graph data (WFTG). Finally, we consider two specific applications of the proposed WFTG: graph data denoising and semi-supervised clustering. Utilizing the sparse representation provided by the WFTG, we propose l 1 -norm based optimization models on graphs for denoising and semi-supervised clustering. On one hand, our numerical results show significant advantage of the WFTG over the spectral graph wavelet transform (SGWT) by [1] for both applications. On the other hand, numerical experiments on two real data sets show that the proposed semi-supervised clustering model using the WFTG is overall competitive with the state-of-the-art methods developed in the literature of high-dimensional data classification, and is superior to some of these methods. 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.