Paper analyzes ADMM convergence for nonconvex Gaussian phase retrieval.
problem Nonconvex optimization in Gaussian phase retrieval.
method Block coordinate descent as ADMM with dual variable fixed.
result Block coordinate descent converges linearly to global minimizer.
New framework uses deep generative priors for robust phase retrieval.
problem Highly ill-posed and non-linear phase retrieval problem.
method Regularization through deep generative priors with gradient descent.
result Effective for random Gaussian and Fourier friendly measurements.
prDeep uses a deep neural network to robustly retrieve phases from noisy data.
problem Noise limits traditional phase retrieval algorithms' performance.
method Regularization-by-denoising framework and convolutional neural network.
result prDeep is robust to noise and can handle various system models.
Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn corresponds to the one of a generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of retrieval capabilities, both at low and high load. We study the paramagnet…
AMP with Gaussian initialization shows weak-recovery threshold for phase retrieval.
problem Phase retrieval with noiseless data.
method Approximate message passing with random initialization.
result Random initialization attains weak-recovery threshold \( \delta_{ ext{weak}} = 1/2 \).
Paper proposes a method for estimating complex low-rank matrices from phase-only measurements.
problem Estimating complex low-rank matrices from magnitude-only measurements.
method A hierarchical prior model with a Gaussian-Wishart distribution is used to promote low-rankness. A variational EM algorithm is developed to solve the problem.
result The proposed method is less sensitive to initialization and performs well with random initialization.
Study on InstaHide's security, linking to phase retrieval problem.
problem Security of InstaHide scheme for private dataset sharing.
method Design of a provable algorithm for private vector recovery.
result Private vectors can be recovered using synthetic vectors and public vectors.
Continuous-time mirror descent solves sparse phase retrieval efficiently.
problem Recovering sparse signals from magnitude-only measurements.
method Continuous-time mirror descent applied to unconstrained empirical risk minimization problem.
result Mirror descent recovers k k k -sparse vectors with minimum non-zero entry order of ∥ x ⋆ ∥ 2 / k \| \mathbf{x}^\star \|_2/\sqrt{k} ∥ x ⋆ ∥ 2 / k from k 2 k^2 k 2 Gaussian measurements. Near-optimal sample complexity for phase retrieval with generative priors.
problem Phase retrieval with magnitude-only measurements and sparse signals.
method Near-optimal sample complexity with i.i.d. Gaussian measurements and generative models.
result O(k log L) samples suffice for phase retrieval with generative priors.
Study shows how anisotropic data affects learning dynamics in phase retrieval.
problem Understanding learning dynamics in phase retrieval with anisotropic Gaussian inputs.
method Developed a tractable reduction to reveal a three-phase trajectory and derived scaling laws.
result Found that anisotropy leads to a three-phase trajectory: fast escape, slow convergence, and spectral-tail learning.
We study Generalised Restricted Boltzmann Machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as Generalised Hopfield models. We underline the role of the r…
Study phase retrieval under misspecified models using generative priors.
problem Estimating signals from phase measurements with model misspecification.
method Two-step approach: spectral initialization followed by iterative refinement.
result Statistical rate of order ( k log L ) ⋅ ( log m ) / m \sqrt{(k\log L)\cdot (\log m)/m} ( k log L ) ⋅ ( log m ) / m under suitable conditions. Gradient descent with random initialization solves phase retrieval problems efficiently.
problem Solving systems of quadratic equations for phase retrieval.
method Gradient descent with random initialization for nonconvex least squares problem.
result Gradient descent achieves near-optimal computational and sample complexities for phase retrieval.
Study on limits of recovering sparse variables from phaseless measurements.
problem Support recovery in phase retrieval model with noisy phaseless measurements.
method Information-theoretic analysis, considering discrete and Gaussian models, Gaussian measurement matrices.
result Sharp thresholds with near-matching constant factors for sparsity and signal-to-noise ratio in various scaling regimes.
SpecGD mitigates misalignment in phase retrieval models with anisotropic inputs.
problem Misalignment during gradient descent in phase retrieval models with anisotropic inputs.
method Spectral gradient descent modifies gradient updates to preserve directional information and remove spike amplification.
result SpecGD removes spike amplification, leading to stable alignment and accelerated noise contraction.
UPR hybrid model improves phase retrieval performance.
problem Recovering signals from phase-less measurements.
method Model-based data-driven deep architecture (UPR).
result UPR shows potential in improving phase retrieval.
This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈ R p x \in \mathbb{R}^p x ∈ R p from noisy quadratic measurements y j = ( a j ′ x ) 2 + ε j y_j = (a_j' x )^2 + ε_j y j = ( a j ′ x ) 2 + ε j , j = 1 , … , m j=1, \ldots, m j = 1 , … , m , with independent sub-exponential noise ε j ε_j ε j . The goals are to understand the effect of the sparsity of x x x on the estimation prec…
Paper studies early-stopped mirror descent for noisy sparse phase retrieval.
problem Recovering a sparse signal from noisy quadratic measurements.
method Early-stopped mirror descent with hyperbolic entropy mirror map.
result Achieves nearly minimax-optimal rate of convergence for k k k -sparse signals. A new deep learning model improves phase retrieval performance.
problem Recovering signals from phaseless measurements.
method Hybrid model-based data-driven deep architecture (Unfolded Phase Retrieval, UPR).
result Significant improvement in phase retrieval performance.
New error bounds for noisy phase retrieval problems using empirical risk minimization.
problem Estimating signals in noisy phase retrieval problems.
method Empirical ℓ 2 \ell_2 ℓ 2 risk minimization (ERM) with new error bounds for different noise patterns. result Established new error bounds for NPR and NGPR, showing improved performance under various noise conditions.
New linear spectral estimators improve phase retrieval accuracy.
problem Recovering vectors from magnitude measurements.
method Linear Spectral Estimators (LSPEs) for phase retrieval.
result LSPEs provide accurate initialization vectors and sharp error bounds.
Phase retrieval requires at least d+o(d) measurements to recover signals with high probability.
problem Recovering signals from quadratic measurements with noisy data.
method Used Gaussian sensing vectors and spectral methods to analyze the minimum number of measurements needed.
result A sharp phase transition occurs at n = d+o(d), where a simple spectral estimator achieves positive correlation.
New algorithms handle phase retrieval with rank d measurements, revealing phase transitions.
problem Phase retrieval with rank d measurements.
method Random duality theory (RDT) and descending phase retrieval algorithms (dPR).
result Minimal sample complexity ratio for dPR's success exhibits phase transitions.
Combines deep learning and iterative methods for robust phase retrieval.
problem Recovering signals from noisy Fourier intensities.
method Regularization-by-denoising combining iterative phase retrieval and deep learning.
result Outperforms other noise-robust phase retrieval algorithms.
Sharp asymptotics derived for phase retrieval and compressed sensing with random generative priors.
problem Phase retrieval and compressed sensing with random measurement matrices.
method Sharp asymptotics derived for optimal performance and polynomial algorithm for random generative priors.
result Compressed phase retrieval becomes tractable with random generative priors, unlike sparse priors.
Guarantees uniform convergence for square-root Lipschitz losses.
problem Uniform convergence guarantees for square-root Lipschitz losses.
method Using Rademacher complexity and square root of scalar loss function Lipschitz constant.
result Generalizes previous results and handles non-smooth loss functions.
Extends phase retrieval methods to handle sensing vector errors.
problem Phase retrieval with errors in sensing vectors.
method Total Least Squares (TLS) framework applied to gradient descent.
result Gradient descent can efficiently solve TLS phase retrieval.
New algorithm solves low-rank phase retrieval with fewer measurements than previously possible.
problem Recovering low-rank matrices from phaseless projections.
method Developed an alternating minimization algorithm (AltMinLowRaP) with provable correctness and geometric convergence.
result The algorithm solves low-rank phase retrieval with m q ≥ C n r 4 log ( 1 / ε ) m q \ge C n r^4 \log(1/ε) m q ≥ C n r 4 log ( 1/ ε ) measurements, achieving ε ε ε accuracy with high probability. Proposes using GANs to solve phase retrieval problems.
problem Solving phase retrieval problems in various contexts.
method Applying conditional GANs with knowledge of measurement process.
result Method provides more robust and detailed solutions.
This paper introduces SRPR for robust phase retrieval with smoothed loss functions.
problem Robust phase retrieval from noisy quadratic measurements with corruptions.
method Smoothed robust phase retrieval (SRPR) using convolution-type smoothed loss functions.
result SRPR has no spurious local solutions and benign landscape under corruptions.
New method uses generative models to improve phase retrieval stability.
problem Improving stability of solutions in phase retrieval problems.
method Unified reconstruction approach using generative models to mitigate overfitting.
result Mitigates overfitting to generative model for varying noise levels.
We consider the problem of recovering a signal x ∗ ∈ R n \mathbf{x}^* \in \mathbf{R}^n x ∗ ∈ R n , from magnitude-only measurements y i = ∣ ⟨ a i , x ∗ ⟩ ∣ y_i = |\left\langle\mathbf{a}_i,\mathbf{x}^*\right\rangle| y i = ∣ ⟨ a i , x ∗ ⟩ ∣ for i = [ m ] i=[m] i = [ m ] . Also called the phase retrieval, this is a fundamental challenge in bio-,astronomical imaging and speech processing. The problem abov…
New algorithm solves phase retrieval with adaptive stopping criteria.
problem Robust phase retrieval problem as nonsmooth, nonconvex optimization.
method Inexact proximal linear algorithm with adaptive stopping criteria.
result Proposed methods are more efficient than existing methods.
New insights on computational limits in analyzing heterogeneous data.
problem Statistical accuracy vs computational tractability in high-dimensional heterogeneous data.
method Oracle-based computational model to establish lower bounds.
result Significant gaps between computationally feasible and classical minimax risks.
New algorithm converges to optimal phase retrieval estimator with misspecified link functions.
problem High-dimensional sparse phase retrieval with incorrect model specification.
method Simple variant of thresholded Wirtinger flow algorithm, linear convergence for optimal accuracy.
result Linear convergence to optimal estimator for a broad family of unknown link functions.
DeepPhaseCut uses neural networks to improve Fourier phase retrieval.
problem Fourier phase retrieval from magnitude data.
method Unsupervised feed-forward neural network with cycleGAN training.
result Outperforms existing methods in Fourier phase retrieval.
Paper proposes a new method for sparse phase retrieval with fewer measurements.
problem Sparse phase retrieval in various fields.
method Stochastic alternating minimizing method (StormSpar) with HTP algorithm.
result The method recovers sparse signals from fewer measurements than existing methods.
Paper tackles sparse phase retrieval with a novel Bayesian approach.
problem Sparse phase retrieval from magnitude-only data.
method Quasi-Bayesian approach using a scaled Student distribution.
result Achieves minimax-optimal convergence rates under sub-exponential noise.
Descending phase retrieval algorithms show a phase transition with increasing sample complexity.
problem Theoretical limits of descending phase retrieval algorithms.
method Utilizing Random duality theory (RDT), the study develops a generic program to characterize algorithm performance.
result As sample complexity increases, the parametric manifold transitions from multi to single funneling points, leading to a phase transition in algorithm success.
We consider the robust phase retrieval problem of recovering the unknown signal from the magnitude-only measurements, where the measurements can be contaminated by both sparse arbitrary corruption and bounded random noise. We propose a new nonconvex algorithm for robust phase retrieval, namely Robust Wirtinger Flow to …
Linear memory stores associations up to a logarithmic scale, but listwise retrieval can handle a quadratic scale.
problem How many key-value associations can a linear memory store?
method Analyzed linear memory models for top-1 and listwise retrieval, proving phase transitions and developing asymptotic theories.
result Linear memory has a logarithmic capacity for top-1 retrieval and a quadratic capacity for listwise retrieval.
Global stability bounds for matrix frames in phase retrieval problems.
problem Phase retrieval for matrix frames in various applications.
method Computable global stability bounds for the quasi-linear analysis map β, using Whitney stratification of positive semidefinite matrices of low rank.
result Novel conditions for a frame to be generalized phase retrievable.
Deep learning tackles low-photon nanoscale holographic phase retrieval.
problem Low-photon imaging challenges at nanoscale.
method Dataset-free deep learning framework with physical model integration.
result Significantly improves signal recovery from higher noise levels.
Efficiently transforms Gaussian data to simulate various target distributions.
problem Generating observations from different target distributions given a single Gaussian observation.
method Designs computationally efficient procedures to approximate target distributions.
result Establishes reduction-based computational lower bounds for high-dimensional statistical models.
We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent …
Paper tackles phase retrieval with robust gradient descent for noisy data.
problem Recover signals from magnitude measurements with noise and corruption.
method Robust gradient descent applied to Wirtinger Flow algorithm.
result Improves algorithm's robustness to heavy-tailed noise and adversarial corruption.
Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of (Gerchberg and Saxton 1972) and (Fienup 1982) is still the popular choice for solving many variants o…
New method uses image registration to recover complex signals from amplitude data.
problem Recovering complex-valued signals from amplitude measurements.
method Indirect registration using LDDMM formalism with exterior calculus.
result Algorithm performs well under various conditions including noise and topology.