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Register a new userBlind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers
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Added by
Stefano Marchesini
Publication date
2018
fields
Electronic Engineering
and research's language is
English
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Ptychography has risen as a reference X-ray imaging technique: it achieves resolutions of one billionth of a meter, macroscopic field of view, or the capability to retrieve chemical or magnetic contrast, among other features. A ptychographyic reconstruction is normally formulated as a blind phase retrieval problem, where both the image (sample) and the probe (illumination) have to be recovered from phaseless measured data. In this article we address a nonlinear least squares model for the blind ptychography problem with constraints on the image and the probe by maximum likelihood estimation of the Poisson noise model. We formulate a variant model that incorporates the information of phaseless measurements of the probe to eliminate possible artifacts. Next, we propose a generalized alternating direction method of multipliers designed for the proposed nonconvex models with convergence guarantee under mild conditions, where their subproblems can be solved by fast element-wise operations. Numerically, the proposed algorithm outperforms state-of-the-art algorithms in both speed and image quality.
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Quantization of the parameters of machine learning models, such as deep neural networks, requires solving constrained optimization problems, where the constraint set is formed by the Cartesian product of many simple discrete sets. For such optimization problems, we study the performance of the Alternating Direction Method of Multipliers for Quantization ($texttt{ADMM-Q}$) algorithm, which is a variant of the widely-used ADMM method applied to our discrete optimization problem. We establish the convergence of the iterates of $texttt{ADMM-Q}$ to certain $textit{stationary points}$. To the best of our knowledge, this is the first analysis of an ADMM-type method for problems with discrete variables/constraints. Based on our theoretical insights, we develop a few variants of $texttt{ADMM-Q}$ that can handle inexact update rules, and have improved performance via the use of soft projection and injecting randomness to the algorithm. We empirically evaluate the efficacy of our proposed approaches.
The alternating direction method of multipliers (ADMM) is one of the most widely used first-order optimisation methods in the literature owing to its simplicity, flexibility and efficiency. Over the years, numerous efforts are made to improve the performance of the method, such as the inertial technique. By studying the geometric properties of ADMM, we discuss the limitations of current inertial accelerated ADMM and then present and analyze an adaptive acceleration scheme for the method. Numerical experiments on problems arising from image processing, statistics and machine learning demonstrate the advantages of the proposed acceleration approach.
In this work, we consider the distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Distributed Alternating Direction Method of Multipliers using Finite-Time Exact Ratio Consensus (D-ADMM-FTERC), to solve the multi-node convex optimization problem, in which every node performs iterative computations and exchanges information with its neighbors. At every iteration of D-ADMM-FTERC, each node solves a local convex optimization problem for the one of the primal variables and utilizes a finite-time exact consensus protocol to obtain the optimal value of the other variable, since the cost function for the second primal variable is not decomposable. Since D-ADMM-FTERC requires to know the upper bound on the number of nodes in the network, we furthermore propose a new algorithm, called Fully D-ADMM Finite-Time Distributed Termination (FD-ADMM-FTDT) algorithm, which does not need any global information. If the individual cost functions are convex and not-necessarily differentiable, the proposed algorithms converge at a rate of O(1/k), where k is the iteration counter. Additionally, if the global objective function is strongly convex and smooth, the proposed algorithms have an approximate R-linear convergence rate. The efficacy of FD-ADMM-FTDT is demonstrated via a distributed L1 regularized logistic regression optimization example. Additionally, comparisons with other state-of-the-art algorithms are provided on large-scale networks showing the superior precision and time-efficient performance of FD-ADMM-FTDT.
In this work, we consider the asynchronous distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Asynchronous Approximate Distributed Alternating Direction Method of Multipliers (AsyAD-ADMM), using finite-time asynchronous approximate ratio consensus, to solve the multi-node convex optimization problem, in which every node performs iterative computations and exchanges information with its neighbors asynchronously. More specifically, at every iteration of AsyAD-ADMM, each node solves a local convex optimization problem for one of the primal variables and utilizes a finite-time asynchronous approximate consensus protocol to obtain the value of the other variable which is close to the optimal value, since the cost function for the second primal variable is not decomposable. If the individual cost functions are convex but not necessarily differentiable, the proposed algorithm converges at a rate of $mathcal{O}(1/k)$, where $k$ is the iteration counter. The efficacy of AsyAD-ADMM is exemplified via a proof-of-concept distributed least-square optimization problem with different performance-influencing factors investigated.
Higher-order tensor canonical polyadic decomposition (CPD) with one or more of the latent factor matrices being columnwisely orthonormal has been well studied in recent years. However, most existing models penalize the noises, if occurring, by employing the least squares loss, which may be sensitive to non-Gaussian noise or outliers, leading to bias estimates of the latent factors. In this paper, based on the maximum a posterior estimation, we derive a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise or outliers. By exploring the half-quadratic property of the model, a new method, which is termed as half-quadratic alternating direction method of multipliers (HQ-ADMM), is proposed to solve the model. Each subproblem involved in HQ-ADMM admits a closed-form solution. Thanks to some nice properties of the Cauchy loss, we show that the whole sequence generated by the algorithm globally converges to a stationary point of the problem under consideration. Numerical experiments on synthetic and real data demonstrate the efficiency and robustness of the proposed model and algorithm.
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