Occupation times of general L\'evy processes

For an arbitrary L\'evy process $X$ which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of $X$ and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of $X$. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications

On the Formation of Elliptical Rings in Disk Galaxies

N-body simulations of galactic collisions are employed to investigate the formation of elliptical rings in disk galaxies. The relative inclination between disk and dwarf galaxies is studied with a fine step of five degrees. It is confirmed that the eccentricity of elliptical ring is linearly proportional to the inclination angle. Deriving from the simulational results, an analytic formula which expresses the eccentricity as a function of time and inclination angle is obtained. This formula shall be useful for the interpretations of the observations of ring systems, and therefore reveals the merging histories of galaxies.Comment: 22 pages, 10 figures, accepted for publication in Ap

Revisiting EmbodiedQA: A Simple Baseline and Beyond

In Embodied Question Answering (EmbodiedQA), an agent interacts with an environment to gather necessary information for answering user questions. Existing works have laid a solid foundation towards solving this interesting problem. But the current performance, especially in navigation, suggests that EmbodiedQA might be too challenging for the contemporary approaches. In this paper, we empirically study this problem and introduce 1) a simple yet effective baseline that achieves promising performance; 2) an easier and practical setting for EmbodiedQA where an agent has a chance to adapt the trained model to a new environment before it actually answers users questions. In this new setting, we randomly place a few objects in new environments, and upgrade the agent policy by a distillation network to retain the generalization ability from the trained model. On the EmbodiedQA v1 benchmark, under the standard setting, our simple baseline achieves very competitive results to the-state-of-the-art; in the new setting, we found the introduced small change in settings yields a notable gain in navigation.Comment: Accepted to IEEE Transactions on Image Processing (TIP

Quantum ergodicity and mixing and their classical limits with quantum kicked rotor

We study the ergodicity and mixing of quantum kicked rotor (QKR) with two distinct approaches. In one approach, we use the definitions of quantum ergodicity and mixing recently proposed in [Phys. Rev. E 94, 022150 (2016)], which involve only eigen-energies (Floquet quasi-energies for QKR). In the other approach, we study ergodicity and mixing with quantum Poincar\`e section, which is plotted with a method that maps a wave function unitarily onto quantum phase space composed of Planck cells. Classical Poincar\`e section can be recovered with the effective Planck constant gradually diminishing. We demonstrate that the two approaches can capture the quantum and classical characteristics of ergodicity and mixing of QKR, and give consistent results with classical model at semiclassical limit. Therefore, we establish a correspondence between quantum ergodicity (mixing) and classical ergodicity (mixing).Comment: 5 figures, 9 page

The Axis-Symmetric Ring Galaxies: AM 0053-353, AM 0147-350, AM 1133-245, AM 1413-243, AM 2302-322, ARP 318, and Head-On Penetrations

Axis-symmetric ring systems can be identified from the new catalog of collisional ring galaxies in Madore et al. (2009). These are O-type-like collisional ring galaxies. Head-on collisions by dwarf galaxies moving along the symmetric axis were performed through N-body simulations to address their origins. It was found that the simulations with smaller initial relative velocities between two galaxies, or the cases with heavier dwarf galaxies, could produce rings with higher density contrasts. There are more than one generation of rings in one collision and the lifetime of any generation of rings is about one dynamical time. It was concluded that head-on penetrations could explain these O-type-like ring galaxies identified from the new catalog in Madore et al. (2009), and the simulated rings resembling the observational O-type-like collisional rings are those at the early stage of one of the ring-generations.Comment: 33 pages, 15 figures, published in ApJ 201

On semi-convergence of generalized skew-Hermitian triangular splitting iteration methods for singular saddle-point problems

Recently, Krukier et al. [Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems, Numer. Linear Algebra Appl. 21 (2014) 152-170] proposed an efficient generalized skew-Hermitian triangular splitting (GSTS) iteration method for nonsingular saddle-point linear systems with strong skew-Hermitian parts. In this work, we further use the GSTS method to solve singular saddle-point problems. The semi-convergence properties of GSTS method are analyzed by using singular value decomposition and Moore-Penrose inverse, under suitable restrictions on the involved iteration parameters. Numerical results are presented to demonstrate the feasibility and efficiency of the GSTS iteration methods, both used as solvers and preconditioners for GMRES method.Comment: 14 page

Spin-Wave Fiber

Spin waves are collective excitations propagating in the magnetic medium with ordered magnetizations. Magnonics, utilizing the spin wave (magnon) as information carrier, is a promising candidate for low-dissipation computation and communication technologies. We discover that, due to the Dzyaloshinskii-Moriya interaction, the scattering behavior of spin wave at a magnetic domain wall follows a generalized Snell's law, where two magnetic domains work as two different mediums. Similar to optical total reflection that occurs at the water-air interfaces, spin waves may experience total reflection at magnetic domain walls when their incident angle larger than a critical value. We design a spin wave fiber using a magnetic domain structure with two domain walls, and demonstrate that such a spin wave fiber can transmit spin waves over long distance by total internal reflections, in analogy to an optical fiber. Our design of spin wave fiber opens up new possibilities in pure magnetic information processing.Comment: 5 pages, 3 figure

Convergence of Contrastive Divergence with Annealed Learning Rate in Exponential Family

In our recent paper, we showed that in exponential family, contrastive divergence (CD) with fixed learning rate will give asymptotically consistent estimates \cite{wu2016convergence}. In this paper, we establish consistency and convergence rate of CD with annealed learning rate $\eta_t$. Specifically, suppose CD-$m$ generates the sequence of parameters $\{\theta_t\}_{t \ge 0}$ using an i.i.d. data sample $\mathbf{X}_1^n \sim p_{\theta^*}$ of size $n$, then $\delta_n(\mathbf{X}_1^n) = \limsup_{t \to \infty} \Vert \sum_{s=t_0}^t \eta_s \theta_s / \sum_{s=t_0}^t \eta_s - \theta^* \Vert$ converges in probability to 0 at a rate of $1/\sqrt[3]{n}$. The number ($m$) of MCMC transitions in CD only affects the coefficient factor of convergence rate. Our proof is not a simple extension of the one in \cite{wu2016convergence}. which depends critically on the fact that $\{\theta_t\}_{t \ge 0}$ is a homogeneous Markov chain conditional on the observed sample $\mathbf{X}_1^n$. Under annealed learning rate, the homogeneous Markov property is not available and we have to develop an alternative approach based on super-martingales. Experiment results of CD on a fully-visible $2\times 2$ Boltzmann Machine are provided to demonstrate our theoretical results

General constraint preconditioning iteration method for singular saddle-point problems

For the singular saddle-point problems with nonsymmetric positive definite $(1,1)$ block, we present a general constraint preconditioning (GCP) iteration method based on a singular constraint preconditioner. Using the properties of the Moore-Penrose inverse, the convergence properties of the GCP iteration method are studied. In particular, for each of the two different choices of the $(1,1)$ block of the singular constraint preconditioner, a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the GCP iteration method. Moreover, the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual (GMRES) and quasi-minimal residual (QMR) methods are also tested

Convergence of Contrastive Divergence Algorithm in Exponential Family

The Contrastive Divergence (CD) algorithm has achieved notable success in training energy-based models including Restricted Boltzmann Machines and played a key role in the emergence of deep learning. The idea of this algorithm is to approximate the intractable term in the exact gradient of the log-likelihood function by using short Markov chain Monte Carlo (MCMC) runs. The approximate gradient is computationally-cheap but biased. Whether and why the CD algorithm provides an asymptotically consistent estimate are still open questions. This paper studies the asymptotic properties of the CD algorithm in canonical exponential families, which are special cases of the energy-based model. Suppose the CD algorithm runs $m$ MCMC transition steps at each iteration $t$ and iteratively generates a sequence of parameter estimates $\{\theta_t\}_{t \ge 0}$ given an i.i.d. data sample $\{X_i\}_{i=1}^n \sim p_{\theta_\star}$. Under conditions which are commonly obeyed by the CD algorithm in practice, we prove the existence of some bounded $m$ such that any limit point of the time average $\left. \sum_{s=0}^{t-1} \theta_s \right/ t$ as $t \to \infty$ is a consistent estimate for the true parameter $\theta_\star$. Our proof is based on the fact that $\{\theta_t\}_{t \ge 0}$ is a homogenous Markov chain conditional on the data sample $\{X_i\}_{i=1}^n$. This chain meets the Foster-Lyapunov drift criterion and converges to a random walk around the Maximum Likelihood Estimate. The range of the random walk shrinks to zero at rate $\mathcal{O}(1/\sqrt[3]{n})$ as the sample size $n \to \infty$