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This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi=|\langle {a}i, {x}\rangle |^{2} , where ai ’s are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adapts the amplitude-based empirical loss function and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai ’s are real valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms. [ABSTRACT FROM PUBLISHER]

We present in this paper two different classes of general multiple-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized can be written as the sum of K convex functions, each of which has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an ε-optimal solution is O((K - 1)L/ε), where L is an upper bound on all of the Lipschitz constants. The algorithms in the second class are accelerated versions of those in the first class, where the complexity result is improved to O(√(K - 1)L/ε) while the computational effort required at each iteration is almost unchanged. To the best of our knowledge, the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction-type methods. Moreover, all algorithms proposed in this paper are parallelizable, which makes them particularly attractive for solving certain large-scale problems. [ABSTRACT FROM AUTHOR]

In the problem of secure message transmission in the public discussion model (SMT-PD), a sender wants to send a message M{\cal S}\in\{0,1\}^{\ell} to a receiver privately and reliably. Sender and receiver are connected by n channels, also known as simple wires, up to t

In this paper, we propose an efficient approach for solving a class of large-scale convex optimization problems. The problem we consider is the minimization of a convex function over a simple (possibly infinite-dimensional) convex set, under the additional constraint Au ∈ T, where A is a linear operator and T is a convex set whose dimension is small compared to the dimension of the feasible region. In our approach, we dualize the linear constraints, solve the resulting dual problem with a purely dual gradient-type method and show how to reconstruct an approximate primal solution. Because the linear constraints have been dualized, the dual objective function typically becomes separable, mid therefore easy to compute. In order to accelerate our scheme, we introduce a novel double smoothing technique that involves regularization of the dual problem to allow the use of a fast gradient method. As a result, we obtain a method with complexity O(1/ε ln 1/ε) gradient iterations, where ε is the desired accuracy for the primal-dual solution. Our approach covers, in particular, optimal control problems with a trajectory governed by a system of linear differential equations, where the additional constraints can, for example, force the trajectory to visit some convex sets at certain moments in time. [ABSTRACT FROM AUTHOR]