# A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval

@article{Wu2020ACM, title={A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval}, author={Fan Wu and Patrick Rebeschini}, journal={ArXiv}, year={2020}, volume={abs/2010.10168} }

We analyze continuous-time mirror descent applied to sparse phase retrieval, which is the problem of recovering sparse signals from a set of magnitude-only measurements. We apply mirror descent to the unconstrained empirical risk minimization problem (batch setting), using the square loss and square measurements. We provide a convergence analysis of the algorithm in this non-convex setting and prove that, with the hypentropy mirror map, mirror descent recovers any $k$-sparse vector $\mathbf{x… Expand

#### Figures from this paper

#### 3 Citations

Implicit Regularization in Matrix Sensing via Mirror Descent

- Mathematics, Computer Science
- ArXiv
- 2021

It is shown that gradient descent with full-rank factorized parametrization is a first-order approximation to mirror descent, in which case an explicit characterization of the implicit bias of gradient flow as a by-product is obtained. Expand

Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity

- Computer Science
- ArXiv
- 2021

The findings highlight the fact that structured noise can induce better generalisation and they help explain the greater performances observed in practice of stochastic gradient descent over gradient descent. Expand

Approximate Message Passing with Spectral Initialization for Generalized Linear Models

- Mathematics, Computer Science
- AISTATS
- 2021

This paper proposes a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP, and yields a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. Expand

#### References

SHOWING 1-10 OF 69 REFERENCES

Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

- Mathematics, Computer Science
- ArXiv
- 2015

A novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the a_j's are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of $x. Expand

Implicit Regularization for Optimal Sparse Recovery

- Computer Science, Mathematics
- NeurIPS
- 2019

We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an… Expand

Algorithmic Regularization in Over-parameterized Matrix Sensing and Neural Networks with Quadratic Activations

- Computer Science, Mathematics
- COLT
- 2018

The gradient descent algorithm provides an implicit regularization effect in the learning of over-parameterized matrix factorization models and one-hidden-layer neural networks with quadratic activations and the results solve the conjecture of Gunasekar et al. Expand

Phase Retrieval via Sparse Wirtinger Flow

- Mathematics, Computer Science
- J. Comput. Appl. Math.
- 2019

Numerical tests demonstrate that SWF performs better than state-of-the-art methods especially when the authors have no priori knowledge about sparsity $k$ and is also robust to the noise. Expand

Phase Retrieval via Wirtinger Flow: Theory and Algorithms

- Computer Science, Mathematics
- IEEE Transactions on Information Theory
- 2015

This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm that is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Expand

Phase Retrieval Using Alternating Minimization

- Computer Science, Mathematics
- IEEE Transactions on Signal Processing
- 2015

This work represents the first theoretical guarantee for alternating minimization (albeit with resampling) for any variant of phase retrieval problems in the non-convex setting. Expand

Sparse Phase Retrieval via Truncated Amplitude Flow

- Mathematics, Computer Science
- IEEE Transactions on Signal Processing
- 2018

A novel algorithm to reconstruct a sparse signal from a small number of magnitude-only measurements, SPARTA is a simple yet effective, scalable, and fast sparse PR solver that is robust against additive noise of bounded support. Expand

Compressive Phase Retrieval via Reweighted Amplitude Flow

- Computer Science, Mathematics
- IEEE Transactions on Signal Processing
- 2018

A new algorithm that is termed compressive reweighted amplitude flow (CRAF) and abbreviated as CRAF, for compressive phase retrieval, which seeks a sparse initial guess via a new spectral procedure and recovers the underlying signal vector exactly with high probability under suitable conditions. Expand

Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval and Matrix Completion

- Computer Science
- ICML
- 2018

Focusing on two statistical estimation problems, i.e. solving random quadratic systems of equations and low-rank matrix completion, it is established that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. Expand

GESPAR: Efficient Phase Retrieval of Sparse Signals

- Mathematics, Computer Science
- IEEE Transactions on Signal Processing
- 2014

This work proposes a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which it refers to as GESPAR: GrEedy Sparse PhAse Retrieval, which does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Expand