# Phase Retrieval Using Alternating Minimization in a Batch Setting

@article{Zhang2018PhaseRU, title={Phase Retrieval Using Alternating Minimization in a Batch Setting}, author={Teng Zhang}, journal={2018 Information Theory and Applications Workshop (ITA)}, year={2018}, pages={1-17} }

This paper considers the problem of phase retrieval, where the goal is to recover a signal $z\in {\mathbb{c}}^n$ from the observations ${y}_{i}=\vert{a}_{i}^\ast{z}\vert, {i}=1,2,\ldots,{m}$. While many algorithms have been proposed, the alternating minimization algorithm has been one of the most commonly used methods, and it is very simple to implement. Current work [26] has proved that when the observation vectors $\{{a}_{i}\}_{i=1}^{m}$ are sampled from a complex Gaussian distribution $N…

## 18 Citations

### Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval

- Computer ScienceMath. Program.
- 2019

This paper provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval, without the need of (i) carefully-designed initialization, (ii) sample splitting, or (iii) sophisticated saddle-point escaping schemes.

### A stochastic alternating minimizing method for sparse phase retrieval

- Computer ScienceArXiv
- 2019

Sparse phase retrieval plays an important role in many fields of applied science and thus attracts lots of attention. In this paper, we propose a \underline{sto}chastic alte\underline{r}nating…

### Learning without the Phase: Regularized PhaseMax Achieves Optimal Sample Complexity

- Computer ScienceNeurIPS
- 2018

A convex optimization problem, where the objective function relies on an initial estimate of the true signal and also includes an additive regularization term to encourage structure, and the new formulation is referred to as regularized PhaseMax.

### Phase retrieval of complex-valued objects via a randomized Kaczmarz method

- Mathematics, Computer ScienceInformation and Inference: A Journal of the IMA
- 2021

The connection between the convergence of the algorithm and the convexity of an objective function is established and it is demonstrated that when the sensing vectors are sampled uniformly from a unit sphere and the number of sensing vectors satisfies $m>O(n\log n)$ as $n, m\rightarrow\infty$, then this algorithm with a good initialization achieves linear convergence to the solution with high probability.

### Phase Retrieval by Alternating Minimization With Random Initialization

- Computer ScienceIEEE Transactions on Information Theory
- 2020

If the phase retrieval problem is considered, it is shown that the classical algorithm of alternating minimization with random initialization succeeds with high probability as theinline-formula, which is a step toward proving the conjecture in, which conjectures that the algorithm succeeds when <inline- formula> <tex-math notation="LaTeX">$m=O(n)$ </tex-Math></inline-Formula>.

### Accelerated Alternating Projections for Robust Principal Component Analysis

- Computer ScienceJ. Mach. Learn. Res.
- 2019

A new algorithm, dubbed accelerated alternating projections, is introduced for robust PCA which significantly improves the computational efficiency of the existing alternating projections proposed in [Netrapalli, Praneeth, et al., 2014] when updating the low rank factor.

### Phase Retrieval via a Modified Null Vector Estimator

- Computer Science2018 10th International Conference on Wireless Communications and Signal Processing (WCSP)
- 2018

Experimental results clearly demonstrate the superiority of the proposed TNI estimator, which not only achieves a lower relative error of initialization but also outperforms the traditional methods in terms of accuracy and noise stability when measurements contaminated with noise.

### Max-Affine Regression: Provable, Tractable, and Near-Optimal Statistical Estimation

- Computer Science, MathematicsArXiv
- 2019

This work analyzes a natural alternating minimization (AM) algorithm for the non-convex least squares objective and shows that the AM algorithm, when initialized suitably, converges with high probability and at a geometric rate to a small ball around the optimal coefficients.

### Sparsity-based Phase Retrieval from Diffractive Optical Imaging

- Computer Science2019 XXII Symposium on Image, Signal Processing and Artificial Vision (STSIVA)
- 2019

Numerical results show that the proposed coding elements are able to solve sparsity-based PR at any optical field under an admissible modulation and show the types of admissible coding elements that best estimate the support.

### A note on Douglas-Rachford, subgradients, and phase retrieval

- Computer ScienceArXiv
- 2019

It is shown that in some cases a generalization of Douglas-Rachford, called relaxed-reflect-reflect (RRR), can be viewed as gradient descent on a certain objective function.

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