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

Chen, Yuxin; Chi, Yuejie; Fan, Jianqing; Ma, Cong

doi:10.1007/s10107-019-01363-6

Cited by 180 publications

(178 citation statements)

References 43 publications

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“…By construction of the variational form, the resulting distribution will be close to |ψ in but slightly perturbed. Adding small random perturbations helps to break symmetries and can, thus, help to improve the training performance [33][34][35]. To create a randomly chosen distribution, we set |ψ in = |0 ⊗3 and initialize the parameters of the variational form following a uniform distribution on [−π, π].…”

Section: A Simulation Studymentioning

confidence: 99%

Quantum Generative Adversarial Networks for learning and loading random distributions

2019

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Quantum algorithms have the potential to outperform their classical counterparts in a variety of tasks. The realization of the advantage often requires the ability to load classical data efficiently into quantum states. However, the best known methods require O (2 n ) gates to load an exact representation of a generic data structure into an n-qubit state. This scaling can easily predominate the complexity of a quantum algorithm and, thereby, impair potential quantum advantage.Our work presents a hybrid quantum-classical algorithm for efficient, approximate quantum state loading. More precisely, we use quantum Generative Adversarial Networks (qGANs) to facilitate efficient learning and loading of generic probability distributions -implicitly given by data samples -into quantum states. Through the interplay of a quantum channel, such as a variational quantum circuit, and a classical neural network, the qGAN can learn a representation of the probability distribution underlying the data samples and load it into a quantum state.The loading requires O (poly (n)) gates and can, thus, enable the use of potentially advantageous quantum algorithms, such as Quantum Amplitude Estimation.We implement the qGAN distribution learning and loading method with Qiskit and test it using a quantum simulation as well as actual quantum processors provided by the IBM Q Experience. Furthermore, we employ quantum simulation to demonstrate the use of the trained quantum channel in a quantum finance application.

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Section: A Simulation Studymentioning

confidence: 99%

Quantum Generative Adversarial Networks for learning and loading random distributions

2019

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“…The log-sigmoid activation function was selected for all the neurons. Several initialization methods for weights and thresholds were tested and results showed that the Gaussian random initialization method [19,20] performed the best. The Levenberg-Marquardt training method [21,22] was used in adjusting the weights and offsets in the backpropagation training process.…”

Section: Methods Developmentmentioning

confidence: 99%

Determination of material optical properties from diffusive reflection light intensity profiles at multiple distances

Liu

Yin

Zhu

et al. 2020

Mater. Res. Express

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Optical absorption and scattering properties are often estimated from the diffusive reflection light intensity at only one distance from the material surface, which often encounters accuracy and convergence issues. In this work, a method was proposed to determine optical properties by using diffusive reflection light intensity profiles at multiple distances, which enhanced data richness as a result of the intensity profiles are linearly independent. In this method, five features of light intensity profiles (contrast, correlation, energy, homogeneity, and second moment) were used to reduce the data dimensions. To demonstrate the effectiveness of the proposed method, Monte Carlo (MC) simulations were used to generate diffusive reflection light intensity profiles with noise at different distances for various combinations of four optical properties (absorption coefficient μ a , scattering coefficient μ s , isotropic coefficient g, and refractive index n). The five profile feature vectors were used as inputs and the four optical parameters were used as outputs to train and test a backpropagation (BP) neural network. The influences of noise levels and the number of diffusive light intensity profiles on parameter estimation accuracy were investigated. The four optical parameters estimated by the BP network were compared with the results estimated by the traditional least squares method, which shows that the proposed method can estimate the optical properties with higher accuracy and better convergence.

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“…[36] reveal that the nonconvex objective (1.5) actually has a benign global geometry: with high probability, it has no bad critical points with m ≥ Ω(n log 3 n) samples 4 . Such a result enables initialization-free nonconvex recovery 5 [42,43].…”

Section: Comparison With Literaturementioning

confidence: 99%

Convolutional Phase Retrieval via Gradient Descent

Zhang

Eldar

et al. 2020

IEEE Trans. Inform. Theory

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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 . This model is motivated by applications such as channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when a is random and the number of observations m is sufficiently large, with high probability x can be efficiently recovered up to a global phase shift using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator. We overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis of alternating minimization methods. ). Most of the work has been conducted when QQ and YZ were with Department of Electrical Engineering and Data Science Institute at Columbia University. An extended abstract of the current work has appeared in NeurIPS'17 [1]. 1 The convolution can be written in matrix-vector form as a x = Caιn→mx, where Ca ∈ R m×m denotes the circulant matrix generated by a and ιn→m is a zero padding operator. In other words, a x = Ax with A ∈ C m×n being a matrix formed by the first n columns of Ca. arXiv:1712.00716v3 [stat.CO] 6 Oct 2019Structured random measurements. The study of structured random measurements in signal processing has quite a long history [44]. For compressed sensing [45], the work [46-48] studied random Fourier measurements, and later [13,14] proved similar results for partial random convolution measurements. However, the study of structured random measurements for phase retrieval is still quite limited. In particular, [49] and [50] studied t-designs and coded diffraction patterns (i.e., random masked Fourier measurements) using semidefinite programming. Recent work studied nonconvex optimization using coded diffraction patterns [34] and STFT measurements [51], both of which minimize a nonconvex objective similar to (1.5). These measurement models are motivated by different applications. For instance, coded diffraction is designed for imaging applications such as X-ray diffraction imaging, STFT can be applied to frequency resolved optical gating [52] and some speech processing tasks [53]. Both of the results show iterative contraction in a region that is at most O(1/ √ n)-close to the optimum. Unfortunately, for both results either the radius of the contraction region is not large enough for initialization to reach, or they require extra artificial technique such as resampling the data. In comparison, the contraction region we show for the random convolutional model is larger O(1/polylog(n)), which is achievable in the initialization stage via the spectral method. For a more detailed review of this subject, we refer the readers to Section 4 of [44]....

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

Cited by 180 publications

References 43 publications

Quantum Generative Adversarial Networks for learning and loading random distributions

Quantum Generative Adversarial Networks for learning and loading random distributions

Determination of material optical properties from diffusive reflection light intensity profiles at multiple distances

Convolutional Phase Retrieval via Gradient Descent

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