We study recovery conditions of weighted 1 minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50% of the (partial) support information is accurate, then weighted 1 minimization is stable and robust under weaker sufficient conditions than the analogous conditions for standard 1 minimization. Moreover, weighted 1 minimization provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals. Index TermsCompressed sensing, weighted 1 minimization, adaptive recovery. I. INTRODUCTIONCompressed sensing (see, e.g., [1]-[3]) is a paradigm for effective acquisition of signals that admit sparse (or approximately sparse) representations in some transform domain. The approach can be used to reliably recover such signals from significantly fewer linear measurements than their ambient dimension.Because a wide range of natural and man-made signals-e.g., audio, natural and seismic images, video, and wideband radio frequency signals-are sparse or approximately sparse in appropriate transform domains, the potential applications of compressed sensing can be immense.

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y = Φx of a k-sparse signal x ∈ R N , where Φ satisfies the restricted isometry property, then the approximate recoveryThe simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth order Σ∆ quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduction of the approximation error by a factor of (m/k) (r−1/2)α for any 0 < α < 1, if m r k(log N ) 1/(1−α) . The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x that satisfy a mild size condition on their supports.

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Consider the recovery of an unknown signal x from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that x is sparse, and that the measurements are of the form sign( a i , x ) ∈ {±1}. Since such measurements give no information on the norm of x, recovery methods typically assume that x 2 = 1. We show that if one allows more generally for quantized affine measurements of the form sign( a i , x + b i ), and if the vectors a i are random, an appropriate choice of the affine shifts b i allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed x in the annulus r ≤ x 2 ≤ R, one may estimate the norm x 2 up to additive error δ from m R 4 r −2 δ −2 such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors x in a Euclidean ball of known radius.

We improve a phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks [27]. The improved algorithm admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees from [27] are improved as a result for the new and more robust reconstruction approach proposed herein. Numerical experiments demonstrate robustness and computational efficiency that outperforms competing approaches on large problems. Finally, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.