A primal-dual algorithm with optimal stepsizes and its application in decentralized consensus optimization

We consider a primal-dual algorithm for minimizing $f(x)+h(Ax)$ with differentiable $f$. The primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP$^2$O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we extend it to solve $f(x)+h\square l(Ax)$ with differentiable $l^*$ and prove its convergence under a weak condition (i.e., under a large dual stepsize). With additional assumptions, we show its linear convergence. In addition, we show that this condition is optimal and can not be weaken. This result recovers the recent proposed positive-indefinite linearized augmented Lagrangian method. Then we consider the application of this primal-dual algorithm in decentralized consensus optimization. We show that EXact firsT-ordeR Algorithm (EXTRA) and Proximal Gradient-EXTRA (PG-EXTRA) can be consider as the primal-dual algorithm applied on a problem in the form of $h\square l(Ax)$. Then, the optimal upper bound of the stepsize for EXTRA/PG-EXTRA is derived. It is larger than the existing work on EXTRA/PG-EXTRA. Furthermore, for the case with strongly convex functions, we proved linear convergence under the same condition for the stepsize.Comment: 19 page

NOON state generation with phonons in acoustic wave resonators assisted by a nitrogen-vacancy-center ensemble

Since the quality factor of an acoustic wave resonator (AWR) reached $10^{11}$, AWRs have been regarded as a good carrier of quantum information. In this paper, we propose a scheme to construct a NOON state with two AWRs assisted by a nitrogen-vacancy-center ensemble (NVE). The two AWRs cross each other vertically, and the NVE is located at the center of the crossing. By considering the decoherence of the system and using resonant interactions between the AWRs and the NVE, and the single-qubit operation of the NVE, a NOON state can be achieved with a fidelity higher than $98.8\%$ when the number of phonons in the AWR is $N \le 3$

Entanglement detection and lower bound of convex-roof extension of negativity

We present a set of inequalities based on mean values of quantum mechanical observables nonlinear entanglement witnesses for bipartite quantum systems. These inequalities give rise to sufficient and necessary conditions for separability of all bipartite pure states and even some mixed states. In terms of these mean values of quantum mechanical observables a measurable lower bound of the convex-roof extension of the negativity is derived.Comment: 8 pages, 1 figur

A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates

This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries

Scheme for conditional generation of photon-added coherent state and optical entangled WW state

We propose a simple scheme to generate an arbitrary photon-added coherent state of a travelling optical field by using only a set of degenerate parametric amplifiers and single-photon detectors. Particularly, when the single-photon-added coherent state (SPACS) is observed by following, e.g., the novel technique of Zavatta \emph{$et al.$} (Science 306, 660 (2004)), we also obtain the generalized optical entangled $W$ state. Finally, a qualitative analysis of possible losses in our scheme is given.Comment: 8 pages, 3 figure

Simultaneous creations of discrete-variable entangle state and single-photon-added coherent state

The single-photon-added coherent state (SPACS), as an intermediate classical-to-purely-quantum state, was first realized recently by Zavatta \emph{$et al.$} (Science 306, 660 (2004)). We show here that the success probability of their SPACS generation can be enhanced by a simple method which leads to simultaneous creations of a discrete-variable entangled state and a SPACS or even a hybrid-variable entangled SPACS in two different channels. The impacts of the input thermal noise are also analyzed.Comment: 6 pages, 1 figur

A Multiphase Image Segmentation Based on Fuzzy Membership Functions and L1-norm Fidelity

In this paper, we propose a variational multiphase image segmentation model based on fuzzy membership functions and L1-norm fidelity. Then we apply the alternating direction method of multipliers to solve an equivalent problem. All the subproblems can be solved efficiently. Specifically, we propose a fast method to calculate the fuzzy median. Experimental results and comparisons show that the L1-norm based method is more robust to outliers such as impulse noise and keeps better contrast than its L2-norm counterpart. Theoretically, we prove the existence of the minimizer and analyze the convergence of the algorithm.Comment: 28 pages, 8 figures, 3 table

Domain Aware Training for Far-field Small-footprint Keyword Spotting

In this paper, we focus on the task of small-footprint keyword spotting under the far-field scenario. Far-field environments are commonly encountered in real-life speech applications, causing severe degradation of performance due to room reverberation and various kinds of noises. Our baseline system is built on the convolutional neural network trained with pooled data of both far-field and close-talking speech. To cope with the distortions, we develop three domain aware training systems, including the domain embedding system, the deep CORAL system, and the multi-task learning system. These methods incorporate domain knowledge into network training and improve the performance of the keyword classifier on far-field conditions. Experimental results show that our proposed methods manage to maintain the performance on the close-talking speech and achieve significant improvement on the far-field test set.Comment: Submitted to INTERSPEECH 202

Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind of method is also $O(\tau^2+h^4)$ under the discrete maximum norm when the space step size is just twice the one of H-OCD method, which accelerates the computational process and makes the result much smoother to some extent. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in time direction. The results of numerical experiments are also consistent with these theoretical analysis.Comment: 30 pages,8 figure

Identifying multi-scale communities in networks by asymptotic surprise

Optimizing statistical measures for community structure is one of the most popular strategies for community detection, but many of them lack the flexibility of resolution and thus are incompatible with multi-scale communities of networks. Here, we further studied a statistical measure of interest for community detection, asymptotic surprise, an asymptotic approximation of surprise. We discussed the critical behaviors of asymptotic surprise in phase transition of community partition theoretically. Then, according to the theoretical analysis, a multi-resolution method based on asymptotic surprise was introduced, which provides an alternative approach to study multi-scale networks, and an improved Louvain algorithm was proposed to optimize the asymptotic surprise more effectively. By a series of experimental tests in various networks, we validated the critical behaviors of the asymptotic surprise further and the effectiveness of the improved Louvain algorithm, displayed its ability to solve the first-type resolution limit and stronger tolerance against the second-type resolution limit, and confirmed its effectiveness of revealing multi-scale community structures in multi-scale networks.Comment: 18 pages, 7 figure