($\ell _1,\ell _2$)-RIP and Projected Back-Projection Reconstruction for Phase-Only Measurements

@article{Feuillen2020ell\_\_,
  title={(\$\ell \_1,\ell \_2\$)-RIP and Projected Back-Projection Reconstruction for Phase-Only Measurements},
  author={Thomas Feuillen and Mike E. Davies and Luc Vandendorpe and Laurent Jacques},
  journal={IEEE Signal Processing Letters},
  year={2020},
  volume={27},
  pages={396-400}
}
This letter analyzes the performances of a simple reconstruction method, namely the Projected Back-Projection (PBP), for estimating the direction of a sparse signal from its phase-only (or amplitude-less) complex Gaussian random measurements, i.e., an extension of one-bit compressive sensing to the complex field. To study the performances of this algorithm, we show that complex Gaussian random matrices respect, with high probability, a variant of the Restricted Isometry Property (RIP) relating… 

Figures from this paper

References

SHOWING 1-10 OF 22 REFERENCES
One-bit compressed sensing with partial Gaussian circulant matrices
TLDR
This paper proves that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property, and establishes stability with respect to approximate sparsity, as well as full vector recovery results.
Sparse Signal Reconstruction from Phase-only Measurements
We demonstrate that the phase of complex linear measurements of signals preserves significant information about the angles between those signals. We provide stable angle embedding guarantees, akin to
Quantized Compressive Sensing with RIP Matrices: The Benefit of Dithering
TLDR
This work shows that, for a scalar and uniform quantization, provided that a uniform random vector, or "random dithering", is added to the compressive measurements of a low-complexity signal, a large class of random matrix constructions known to respect the restricted isometry property (RIP) are made "compatible" with this quantizer.
1-Bit compressive sensing
TLDR
This paper reformulates the problem by treating the 1-bit measurements as sign constraints and further constraining the optimization to recover a signal on the unit sphere, and demonstrates that this approach performs significantly better compared to the classical compressive sensing reconstruction methods, even as the signal becomes less sparse and as the number of measurements increases.
Dequantizing Compressed Sensing: When Oversampling and Non-Gaussian Constraints Combine
TLDR
A new class of convex optimization programs, or decoders, coined Basis Pursuit DeQuantizer of moment p (BPDQp), that model the quantization distortion more faithfully than the commonly used Basis pursuit DeNoise (B PDN) program are presented.
Iterative Hard Thresholding for Compressed Sensing
Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors
TLDR
This paper investigates an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement, and introduces the binary iterative hard thresholding algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.
Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements
TLDR
This paper shows that if an rth-order ΣΔ (Sigma–Delta) 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 reduced approximation error of the order δ(k/m)(r−1/2)α for any 0<α<1, if m≳r,αk(logN)1/(1−α).
The Generalized Lasso With Non-Linear Observations
TLDR
The first theoretical accuracy guarantee for 1-b compressed sensing with unknown covariance matrix of the measurement vectors is given, and the single-index model of non-linearity is considered, allowing the non- linearity to be discontinuous, not one-to-one and even unknown.
A Simple Proof of the Restricted Isometry Property for Random Matrices
Abstract We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main
...
...