Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem from an optimal number of measurements and present a partial result in this direction for the general rank problem.
more »
« less
Simultaneous Blind Deconvolution and Phase Retrieval with Tensor Iterative Hard Thresholding
Blind deconvolution and phase retrieval are both fundamental problems with a growing interest in signal processing and communications. In this work, we consider the task of simultaneous blind deconvolution and phase retrieval. We show that this non-linear problem can be reformulated as a low-rank tensor recovery problem and propose an algorithm named TIHT-BDPR to recover the unknown parameters. We include a series of numerical simulations to illustrate the effectiveness of our proposed algorithm.
more »
« less
- Award ID(s):
- 1704204
- PAR ID:
- 10099568
- Date Published:
- Journal Name:
- ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- Page Range / eLocation ID:
- 2977 to 2981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)In this work, we analyze the convergence of constant modulus algorithm (CMA) in blindly recovering multiple signals to facilitate grant-free wireless access. The CMA typically solves a non-convex problem by utilizing stochastic gradient descent. The iterative convergence of CMA can be affected by additive channel noise and finite number of samples, which is a problem not fully investigated previously. We point out the strong similarity between CMA and the Wirtinger Flow (WF) algorithm originally proposed for Phase retrieval. In light of the convergence proof of WF under limited data samples, we adopt the WF algorithm to implement CMA-based blind signal recovery. We generalize the convergence analysis of WF in the context of CMA-based blind signal recovery. Numerical simulation results also corroborate the analysis.more » « less
-
This paper deals with the problem of blind identification of a graph filter and its sparse input signal, thus broadening the scope of classical blind deconvolution of temporal and spatial signals to irregular graph domains. While the observations are bilinear functions of the unknowns, a mild requirement on invertibility of the filter enables an efficient convex formulation, without relying on matrix lifting that can hinder applicability to large graphs. On top of scaling, it is argued that (non-cyclic) permutation ambiguities may arise with some particular graphs. Deterministic sufficient conditions under which the proposed convex relaxation can exactly recover the unknowns are stated, along with those guaranteeing identifiability under the Bernoulli-Gaussian model for the inputs. Numerical tests with synthetic and real-world networks illustrate the merits of the proposed algorithm, as well as the benefits of leveraging multiple signals to aid the (blind) localization of sources of diffusion.more » « less
-
Phase retrieval deals with the recovery of complex-or real-valued signals from magnitude measurements. As shown recently, the method PhaseMax enables phase retrieval via convex optimization and without lifting the problem to a higher dimension. To succeed, PhaseMax requires an initial guess of the solution, which can be calculated via spectral initializers. In this paper, we show that with the availability of an initial guess, phase retrieval can be carried out with an ever simpler, linear procedure. Our algorithm, called PhaseLin, is the linear estimator that minimizes the mean squared error (MSE) when applied to the magnitude measurements. The linear nature of PhaseLin enables an exact and nonasymptotic MSE analysis for arbitrary measurement matrices. We furthermore demonstrate that by iteratively using PhaseLin, one arrives at an efficient phase retrieval algorithm that performs on par with existing convex and nonconvex methods on synthetic and real-world data.more » « less
-
Typically, inversion algorithms assume that a forward model, which relates a source to its resulting measurements, is known and fixed. Using collected indirect measurements and the forward model, the goal becomes to recover the source. When the forward model is unknown, or imperfect, artifacts due to model mismatch occur in the recovery of the source. In this paper, we study the problem of blind inversion: solving an inverse problem with unknown or imperfect knowledge of the forward model parameters. We propose DeepGEM, a variational Expectation-Maximization (EM) framework that can be used to solve for the unknown parameters of the forward model in an unsupervised manner. DeepGEM makes use of a normalizing flow generative network to efficiently capture complex posterior distributions, which leads to more accurate evaluation of the source's posterior distribution used in EM. We showcase the effectiveness of our DeepGEM approach by achieving strong performance on the challenging problem of blind seismic tomography, where we significantly outperform the standard method used in seismology. We also demonstrate the generality of DeepGEM by applying it to a simple case of blind deconvolution.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICASSP.2019.8683575
