Filters








9 Hits in 3.1 sec

An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax [article]

Paul Hand, Vladislav Voroninski
2016 arXiv   pre-print
Very recently, a convex formulation called PhaseMax has been discovered, and it has been proven to achieve phase retrieval via linear programming in the natural parameter space under optimal sample complexity  ...  The phase retrieval problem has garnered significant attention since the development of the PhaseLift algorithm, which is a convex program that operates in a lifted space of matrices.  ...  To the surprise of the community, a recent successful formulation for phase retrieval called Phase-Max, independently developed in [7, 1] , is convex and operates in the natural n-dimensional parameter  ... 
arXiv:1611.03935v1 fatcat:7c2x3fa6orfafonzttd37n2bzm

Corruption Robust Phase Retrieval via Linear Programming [article]

Paul Hand, Vladislav Voroninski
2016 arXiv   pre-print
This linear programming formulation, which we call RobustPhaseMax, operates in the natural parameter space, and our proofs rely on a direct analysis of the optimality conditions using concentration inequalities  ...  We consider the problem of phase retrieval from corrupted magnitude observations.  ...  Acknowledgements PH acknowledges funding by the grant NSF DMS-1464525.  ... 
arXiv:1612.03547v1 fatcat:yvnkas7l5fgddo63xturu7wavy

Convex Phase Retrieval without Lifting via PhaseMax

Tom Goldstein, Christoph Studer
2017 International Conference on Machine Learning  
We study a new type of convex relaxation for phase retrieval problems, called PhaseMax, that convexifies the underlying problem without lifting.  ...  We compare our approach to other phase retrieval methods and demonstrate that our theory accurately predicts the success of PhaseMax.  ...  Acknowledgments The work of T.  ... 
dblp:conf/icml/GoldsteinS17 fatcat:3auw4zxpwjbv7e6otjwpeaqspq

Phase Retrieval Under a Generative Prior [article]

Paul Hand, Oscar Leong, Vladislav Voroninski
2018 arXiv   pre-print
While successful in the realm of linear inverse problems, such ℓ_1 methods have encountered possibly fundamental limitations, as no computationally efficient algorithm for phase retrieval of a k-sparse  ...  In this paper, we propose a novel framework for phase retrieval by 1) modeling natural signals as being in the range of a deep generative neural network G : R^k →R^n and 2) enforcing this prior directly  ...  Other convex methods include PhaseCut [31] , an SDP approach, and linear programming algorithms such as PhaseMax [13] Non-convex methods encompass alternating minimization approaches such as the original  ... 
arXiv:1807.04261v1 fatcat:r7nkdt3b6vcxnnezktsrynjgo4

Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution [article]

Cong Ma, Kaizheng Wang, Yuejie Chi, Yuxin Chen
2019 arXiv   pre-print
Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence  ...  In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument  ...  Acknowledgements The work of Y. Chi is supported in part by the grants AFOSR FA9550-15-1-0205, ONR N00014-15-1-2387, NSF CCF-1527456, ECCS-1650449 and CCF-1704245. Y.  ... 
arXiv:1711.10467v3 fatcat:qmqqujn55nhtbncwcy3c7ka66m

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

Cong Ma, Kaizheng Wang, Yuejie Chi, Yuxin Chen
2019 Foundations of Computational Mathematics  
Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g., trimming, regularized cost, projection) in order to guarantee fast convergence  ...  In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument  ...  Another convex program PhaseMax [5, 41, 50, 53] operates in the natural parameter space via linear programming, provided that an anchor vector is available.  ... 
doi:10.1007/s10208-019-09429-9 fatcat:n3ke3ixyorejhcqehzlt3uxmke

Impact of the Sensing Spectrum on Signal Recovery in Generalized Linear Models [article]

Junjie Ma, Ji Xu, Arian Maleki
2021 arXiv   pre-print
Based on our framework, we are able to show that for instance, in phase-retrieval problems, matrices with spikier spectrums are better for EP, while in 1-bit compressed sensing problems, less spiky (flatter  ...  We define a notion for the spikiness of the spectrum of 𝐀 and show the importance of this measure in the performance of the EP.  ...  Phasemax: Convex phase retrieval via basis pursuit. IEEE Transactions on Information Theory, 64(4):2675–2689, April 2018. [29] D. Guo, Y. Wu, S. S. Shitz, and S. Verdu.  ... 
arXiv:2111.03237v2 fatcat:obamdwrt2benfmtwijiifnngwq

On the Cryptographic Hardness of Learning Single Periodic Neurons [article]

Min Jae Song, Ilias Zadik, Joan Bruna
2021 arXiv   pre-print
Furthermore, in the absence of noise, this algorithm can be directly applied to solve CLWE detection (Bruna et al.'21) and phase retrieval with an optimal sample complexity of d+1 samples.  ...  Our core hard family of functions, which are well-approximated by one-layer neural networks, take the general form of a univariate periodic function applied to an affine projection of the data.  ...  We also thank Daniel Hsu for pointing out the relevant prior work [And+17] after an initial version of our manuscript was posted online. MS and JB are partially supported by the Alfred P.  ... 
arXiv:2106.10744v2 fatcat:ygdqhxpkqbgyhlvkfuebjg6k5u

Convolutional Phase Retrieval via Gradient Descent [article]

Qing Qu, Yuqian Zhang, Yonina C. Eldar, John Wright
2019 arXiv   pre-print
We study the convolutional phase retrieval problem, of recovering an unknown signal x∈C^n from m measurements consisting of the magnitude of its cyclic convolution with a given kernel a∈C^m.  ...  The main challenge is coping with dependencies in the measurement operator.  ...  Quite recently, [30] [31] [32] reveal that the problem can also be solved in the natural parameter space via linear programming.  ... 
arXiv:1712.00716v3 fatcat:3hu5arfwx5dvjeze7balnyz2uq