Recovery analysis for weighted mixed ℓ2/ℓp\ell_2/\ell_p minimization with 0<p≤10<p\leq 1

We study the recovery conditions of weighted mixed $\ell_2/\ell_p\,(0<p\leq 1)$ minimization for block sparse signal reconstruction from compressed measurements when partial block support information is available. We show that the block $p$-restricted isometry property (RIP) can ensure the robust recovery. Moreover, we present the sufficient and necessary condition for the recovery by using weighted block $p$-null space property. The relationship between the block $p$-RIP and the weighted block $p$-null space property has been established. Finally, we illustrate our results with a series of numerical experiments

Support Driven Wavelet Frame-based Image Deblurring

The wavelet frame systems have been playing an active role in image restoration and many other image processing fields over the past decades, owing to the good capability of sparsely approximating piece-wise smooth functions such as images. In this paper, we propose a novel wavelet frame based sparse recovery model called \textit{Support Driven Sparse Regularization} (SDSR) for image deblurring, where the partial support information of frame coefficients is attained via a self-learning strategy and exploited via the proposed truncated $\ell_0$ regularization. Moreover, the state-of-the-art image restoration methods can be naturally incorporated into our proposed wavelet frame based sparse recovery framework. In particular, in order to achieve reliable support estimation of the frame coefficients, we make use of the state-of-the-art image restoration result such as that from the IDD-BM3D method as the initial reference image for support estimation. Our extensive experimental results have shown convincing improvements over existing state-of-the-art deblurring methods

Robust width: A characterization of uniformly stable and robust compressed sensing

Compressed sensing seeks to invert an underdetermined linear system by exploiting additional knowledge of the true solution. Over the last decade, several instances of compressed sensing have been studied for various applications, and for each instance, reconstruction guarantees are available provided the sensing operator satisfies certain sufficient conditions. In this paper, we completely characterize the sensing operators which allow uniformly stable and robust reconstruction by convex optimization for many of these instances. The characterized sensing operators satisfy a new property we call the robust width property, which simultaneously captures notions of widths from approximation theory and of restricted eigenvalues from statistical regression. We provide a geometric interpretation of this property, we discuss its relationship with the restricted isometry property, and we apply techniques from geometric functional analysis to find random matrices which satisfy the property with high probability.Comment: 24 page

Polynomial approximation via compressed sensing of high-dimensional functions on lower sets

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted $\ell_1$ minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted $\ell_1$ minimization strategy.Comment: 33 pages, 3 figure

Correcting for unknown errors in sparse high-dimensional function approximation

We consider sparsity-based techniques for the approximation of high-dimensional functions from random pointwise evaluations. To date, almost all the works published in this field contain some a priori assumptions about the error corrupting the samples that are hard to verify in practice. In this paper, we instead focus on the scenario where the error is unknown. We study the performance of four sparsity-promoting optimization problems: weighted quadratically-constrained basis pursuit, weighted LASSO, weighted square-root LASSO, and weighted LAD-LASSO. From the theoretical perspective, we prove uniform recovery guarantees for these decoders, deriving recipes for the optimal choice of the respective tuning parameters. On the numerical side, we compare them in the pure function approximation case and in applications to uncertainty quantification of ODEs and PDEs with random inputs. Our main conclusion is that the lesser-known square-root LASSO is better suited for high-dimensional approximation than the other procedures in the case of bounded noise, since it avoids (both theoretically and numerically) the need for parameter tuning