A strong restricted isometry property, with an application to phaseless compressed sensing

  title={A strong restricted isometry property, with an application to phaseless compressed sensing},
  author={Vladislav Voroninski and Zhiqiang Xu},
Stable Signal Recovery from Phaseless Measurements
This paper shows that m={\mathcal {O}}(k\log (N/k) measurements are enough to guarantee the phaseless instance-optimality for the ℓ1 minimization to recover k-sparse signals stably provided the measurement matrix A satisfies the strong RIP property.
On the Erasure Robustness Property of Random Matrices
  • Ran Lu
  • Computer Science, Mathematics
  • 2017
The erasure robustness of $\pm 1$ random matrices is studied to show that with overwhelming probability the SRIP will still hold and the analysis will also lead to the robust version of the Johnson-Lindenstrauss Lemma for $\pm1$ matrices.
On the strong restricted isometry property of Bernoulli random matrices
  • Ran Lu
  • Mathematics, Computer Science
    J. Approx. Theory
  • 2019
Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Space
This work establishes that when provided with an initialization that correlates with an arbitrary $k-sparse $n$-vector, SparsePhaseMax recovers this vector up to global sign with high probability from $O(k \log \frac{n}{k})$ magnitude measurements against i.i.d. Gaussian random vectors.
Hadamard Wirtinger Flow for Sparse Phase Retrieval
The sample complexity performance of gradient descent with Hadamard parametrization is studied with HWF and it is proved that a single step of HWF is able to recover the support from $\mathcal{O}(k(x^*_{max})^{-2} \log n)$ samples.
Stable recovery of weighted sparse signals from phaseless measurements via weighted l1 minimization
  • Haiye Huo
  • Computer Science
    Mathematical Methods in the Applied Sciences
  • 2022
This paper proves that the weighted null space property (WNSP) is a sufficient and necessary condition for the success of weighted l1 minimization for weighted k‐sparse phase retrievable and shows that if a measurement matrix satisfies the strong weighted restricted isometry property (SWRIP), then the original signal can be stably recovered from the phaseless measurements.
Corruption Robust Phase Retrieval via Linear Programming
It is shown that a fixed x_0 can be recovered exactly from corrupted magnitude measurements with high probability for $m = O(n)$, where $a_i \in \mathbb{R}^n$ are i.i.d standard Gaussian and $\eta \in £m$ has fixed sparse support.
Robustness properties of dimensionality reduction with Gaussian random matrices
The robustness property against erasure for the almost norm preservation property of Gaussian random matrices is studied by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate and the strong restricted isometry property with the almost optimal restricted isometric property (RIP) constants is established.
A Geometric Analysis of Phase Retrieval
It is proved that when the measurement vectors are generic, with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal, up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point.


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
Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices
This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
It is shown that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques, and it is proved that the methodology is robust vis‐à‐vis additive noise.
Compressive Phase Retrieval via Generalized Approximate Message Passing
A novel, probabilistic approach to compressive phase retrieval based on the generalized approximate message passing (GAMP) algorithm, and suggests that PR-GAMP has a superior phase transition and orders-of-magnitude faster runtimes as the sparsity and problem dimensions increase.
Sparse Signal Recovery from Quadratic Measurements via Convex Programming
It is proved that if there exists a sparse solution x, i.e., at most k components of x are non-zero, then by solving a convex optimization program, the authors can solve for x up to a multiplicative constant with high probability, provided that k <= O((m/log n)^(1/2)).
New and Improved Johnson-Lindenstrauss Embeddings via the Restricted Isometry Property
The results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices and improve the recent bound m = O(delta^(-4) log(p) log^4(N)) appearing in Ailon and Liberty, which is optimal up to the logarithmic factors in N.
Compressed sensing and best k-term approximation
The typical paradigm for obtaining a compressed version of a discrete signal represented by a vector x ∈ R is to choose an appropriate basis, compute the coefficients of x in this basis, and then
Compressed Sensing
This report explains the ideas of compressed sensing to the reader and gives a (highly incomplete) overview of the work done in the field and suggests that proofs involving matrices can be shortened by 50% if one throws the matrices out.