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.
Due to the similarities between the DFT and the WHT, we give a brief account of previous work on reducing the sample and computational complexity of obtaining a @math -sparse @math -point DFT. The most related research thread in the literature is the computation of sparse DFT using theoretical computer science techniques such as sketching and hashing (see @cite_13 @cite_17 @cite_26 @cite_16 @cite_20 ). Most of these algorithms aim at minimizing the approximation error of the DFT coefficients using an @math -norm metric instead of exact support recovery (i.e., @math -norm).
@cite_26 @cite_16 @cite_13 @cite_20 @cite_17
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In recent years, sparse coding has been widely used in many applications ranging from image processing to pattern recognition. Most existing sparse coding based applications require solving a class of challenging non-smooth and non-convex optimization problems. Despite the fact that many numerical methods have been developed for solving these problems, it remains an open problem to find a numerical method which is not only empirically fast, but also has mathematically guaranteed strong convergence. In this paper, we propose an alternating iteration scheme for solving such problems. A rigorous convergence analysis shows that the proposed method satisfies the global convergence property: the whole sequence of iterates is convergent and converges to a critical point. Besides the theoretical soundness, the practical benefit of the proposed method is validated in applications including image restoration and recognition. Experiments show that the proposed method achieves similar results with less computation when compared to widely used methods such as K-SVD. We study the recovery of sparse vectors from subsampled random convolutions via @math -minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity @math is small enough, i.e., @math , we show that @math measurements are sufficient to recover @math -sparse vectors in dimension @math with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If @math is larger, then essentially @math measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques which should be of independent interest. 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. We consider solving the @math -regularized least-squares ( @math -LS) problem in the context of sparse recovery for applications such as compressed sensing. The standard proximal gradient method, also known as iterative soft-thresholding when applied to this problem, has low computational cost per iteration but a rather slow convergence rate. Nevertheless, when the solution is sparse, it often exhibits fast linear convergence in the final stage. We exploit the local linear convergence using a homotopy continuation strategy, i.e., we solve the @math -LS problem for a sequence of decreasing values of the regularization parameter, and use an approximate solution at the end of each stage to warm start the next stage. Although similar strategies have been studied in the literature, there have been no theoretical analysis of their global iteration complexity. This paper shows that under suitable assumptions for sparse recovery, the proposed homotopy strategy ensures that all iterates along the homotopy sol...