• Corpus ID: 139105193

Simultaneous Phase Retrieval and Blind Deconvolution via Convex Programming

@article{Ahmed2019SimultaneousPR,
  title={Simultaneous Phase Retrieval and Blind Deconvolution via Convex Programming},
  author={Ali Ahmed and Alireza Aghasi and Paul Hand},
  journal={J. Mach. Learn. Res.},
  year={2019},
  volume={20},
  pages={157:1-157:28}
}
We consider the task of recovering two real or complex $m$-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a nontrivial convex relaxation of the bilinear measurements from convolution. We prove that if the two signals belong to known random subspaces of dimensions $k$ and $n$, then they can be recovered up to the inherent scaling ambiguity with $m \gg (k+n) \log^2… 
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References

SHOWING 1-10 OF 36 REFERENCES

Blind Deconvolution Using Convex Programming

It is proved that, for “generic” signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L, and it is shown that if x is drawn from a random sub space of dimension N, and w is a vector in a subspace of dimension K whose basis vectors are spread out in the frequency domain, then nuclear norm minimization recovers wx* without error.

Convex inversion of the entrywise product of real signals with known signs

The convex program BranchHull is introduced, which is posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved on the bilinear inverse problem of recovering two vectors in RL from their entrywise product.

PhaseMax: Convex Phase Retrieval via Basis Pursuit

This work formulate phase retrieval as a convex optimization problem, which it is shown that the dual problem to PhaseMax is basis pursuit, which implies that the phase retrieval can be performed using algorithms initially designed for sparse signal recovery.

Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation

A flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal to avoid the prohibitive computational cost associated with "lifting" and semidefinite programming and compete with recently developed non-convex techniques for phase retrieval.

BranchHull: Convex bilinear inversion from the entrywise product of signals with known signs

Geometry and Symmetry in Short-and-Sparse Deconvolution

A method based on nonconvex optimization is proposed, which under certain conditions recovers the target short and sparse signals, up to a signed shift symmetry which is intrinsic to this model.

A convex program for bilinear inversion of sparse vectors

The introduced $\ell_1$-BranchHull, which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved, and an ADMM implementation of these variants are provided.

Anchored Regression: Solving Random Convex Equations via Convex Programming

We consider the problem of estimating a solution to (random) systems of equations that involve convex nonlinearities which has applications in machine learning and signal processing. Conventional…

Regularization and the small-ball method I: sparse recovery

We obtain bounds on estimation error rates for regularization procedures of the form \begin{equation*} \hat f \in {\rm argmin}_{f\in F}\left(\frac{1}{N}\sum_{i=1}^N\left(Y_i-f(X_i)\right)^2+\lambda…

A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

A nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form that replaces the symmetric, positive semideFinite variable X with a rectangular variable R according to the factorization X=RRT.