Linearizability conditions of quasi-cubic systems

In this paper we study the linearizability problem of the two-dimensional complex quasi-cubic system $\dot{z}=z+(zw)^{d}(a_{30}z^{3}+a_{21}z^{2}w+a_{12}zw^2+a_{03}w^{3}),~\dot{w}=-w-(zw)^{d}(b_{30}w^{3}+b_{21}w^{2}z+b_{12}wz^2+b_{03}z^{3})$, where $z, w, a_{ij}, b_{ij}\in \mathbb{C}$ and $d$ is a real number. We find a transformation to change the quasi-cubic system into an equivalent quintic system and then obtain the necessary and sufficient linearizability conditions by the Darboux linearization method or by proving the existence of linearizing transformations

High-Dimensional Non-Gaussian Data Clustering using Variational Learning of Mixture Models

Clustering has been the topic of extensive research in the past. The main concern is to automatically divide a given data set into different clusters such that vectors of the same cluster are as similar as possible and vectors of different clusters are as different as possible. Finite mixture models have been widely used for clustering since they have the advantages of being able to integrate prior knowledge about the data and to address the problem of unsupervised learning in a formal way. A crucial starting point when adopting mixture models is the choice of the components densities. In this context, the well-known Gaussian distribution has been widely used. However, the deployment of the Gaussian mixture implies implicitly clustering based on the minimization of Euclidean distortions which may yield to poor results in several real applications where the per-components densities are not Gaussian. Recent works have shown that other models such as the Dirichlet, generalized Dirichlet and Beta-Liouville mixtures may provide better clustering results in applications containing non-Gaussian data, especially those involving proportional data (or normalized histograms) which are naturally generated by many applications. Two other challenging aspects that should also be addressed when considering mixture models are: how to determine the model's complexity (i.e. the number of mixture components) and how to estimate the model's parameters. Fortunately, both problems can be tackled simultaneously within a principled elegant learning framework namely variational inference. The main idea of variational inference is to approximate the model posterior distribution by minimizing the Kullback-Leibler divergence between the exact (or true) posterior and an approximating distribution. Recently, variational inference has provided good generalization performance and computational tractability in many applications including learning mixture models. In this thesis, we propose several approaches for high-dimensional non-Gaussian data clustering based on various mixture models such as Dirichlet, generalized Dirichlet and Beta-Liouville. These mixture models are learned using variational inference which main advantages are computational efficiency and guaranteed convergence. More specifically, our contributions are four-fold. Firstly, we develop a variational inference algorithm for learning the finite Dirichlet mixture model, where model parameters and the model complexity can be determined automatically and simultaneously as part of the Bayesian inference procedure; Secondly, an unsupervised feature selection scheme is integrated with finite generalized Dirichlet mixture model for clustering high-dimensional non-Gaussian data; Thirdly, we extend the proposed finite generalized mixture model to the infinite case using a nonparametric Bayesian framework known as Dirichlet process, so that the difficulty of choosing the appropriate number of clusters is sidestepped by assuming that there are an infinite number of mixture components; Finally, we propose an online learning framework to learn a Dirichlet process mixture of Beta-Liouville distributions (i.e. an infinite Beta-Liouville mixture model), which is more suitable when dealing with sequential or large scale data in contrast to batch learning algorithm. The effectiveness of our approaches is evaluated using both synthetic and real-life challenging applications such as image databases categorization, anomaly intrusion detection, human action videos categorization, image annotation, facial expression recognition, behavior recognition, and dynamic textures clustering

Novel approaches for generating video textures

Video texture, a new type of medium, can produce a new video with a continuously varying stream of images from a recorded video. It is synthesized by reordering the input video frames in a way which can be played without any visual discontinuity. However, video texture still experiences few unappealing drawbacks. For instance, video texture techniques can only generate new videos by simply rearranging the order of frames in original videos. Therefore, all the individual frames are the same as before and the result would suffer from "dead-ends" if the current frame could not discover similar frames to make a transition. In this thesis, we propose several new approaches for synthesizing video textures. These approaches adopt dimensionality reduction and regression techniques to generate video textures. Not only the frames in the resulted video textures are new, but also the "Dead end" problem is avoided. First, we have extended die work of applying principal components analysis (PCA) and autoregressive (AR) process to generate video textures by replacing PCA with five other dimension reduction techniques. Based on our experiments, using these dimensionality reduction techniques has improved the quality of video textures compared with extraction of frame signatures using PCA. The synthesized video textures may contain similar motions as the input video and will never be repeated exactly. All frames synthesized have never appeared before. We also propose a new approach for generating video textures using probabilistic principal components analysis (PPCA) and Gaussian process dynamical model (GPDM). GPDM is a nonparametric model for learning high-dimensional nonlinear dynamical data sets. We adopt PPCA and GPDM on several movie clips to synthesize video textures which contain frames that never appeared before and with similar motions as original videos. Furthermore, we have proposed two ways of generating real-time video textures by applying the incremental Isomap and incremental Spati04emporal Isomap (IST-Isomap). Both approaches can produce good real-time video texture results. In particular, IST-Isomap, that we propose, is more suitable for sparse video data (e.g. cartoon

Anti-periodic solutions for a class of third-order nonlinear differential equations with a deviating argument

In this paper, we study a class of third-order nonlinear differential equations with a deviating argument and establish some sufficient conditions for the existence and exponential stability of anti-periodic solutions of the equation. These conditions are new and complement to previously known results