In this paper, we develop a parameter estimation method for factorially parametrized models such as Factorial Gaussian Mixture Model and Factorial Hidden Markov Model. Our contributions are two-fold. First, we show that the emission matrix of the standard Factorial Model is unidentifiable even if the true assignment matrix is known. Secondly, we address the issue of identifiability by making a one component sharing assumption and derive a parameter learning algorithm for this case. Our approach is based on a dictionary learning problem of the form $X = O R$, where the goal is to learn the dictionary $O$ given the data matrix $X$. We argue that due to the specific structure of the activation matrix $R$ in the shared component factorial mixture model, and an incoherence assumption on the shared component, it is possible to extract the columns of the $O$ matrix without the need for alternating between the estimation of $O$ and $R$.

Matrix rank minimizing subject to affine constraints arises in many application areas, ranging from signal processing to machine learning. Nuclear norm is a convex relaxation for this problem which can recover the rank exactly under some restricted and theoretically interesting conditions. However, for many real-world applications, nuclear norm approximation to the rank function can only produce a result far from the optimum. To seek a solution of higher accuracy than the nuclear norm, in this paper, we propose a rank approximation based on Logarithm-Determinant. We consider using this rank approximation for subspace clustering application. Our framework can model different kinds of errors and noise. Effective optimization strategy is developed with theoretical guarantee to converge to a stationary point. The proposed method gives promising results on face clustering and motion segmentation tasks compared to the state-of-the-art subspace clustering algorithms.

We present a novel approach for fully non-stationary Gaussian process regression (GPR), where all three key parameters -- noise variance, signal variance and lengthscale -- can be simultaneously input-dependent. We develop gradient-based inference methods to learn the unknown function and the non-stationary model parameters, without requiring any model approximations. We propose to infer full parameter posterior with Hamiltonian Monte Carlo (HMC), which conveniently extends the analytical gradient-based GPR learning by guiding the sampling with model gradients. We also learn the MAP solution from the posterior by gradient ascent. In experiments on several synthetic datasets and in modelling of temporal gene expression, the nonstationary GPR is shown to be necessary for modeling realistic input-dependent dynamics, while it performs comparably to conventional stationary or previous non-stationary GPR models otherwise.

We present a Bayesian non-negative tensor factorization model for count-valued tensor data, and develop scalable inference algorithms (both batch and online) for dealing with massive tensors. Our generative model can handle overdispersed counts as well as infer the rank of the decomposition. Moreover, leveraging a reparameterization of the Poisson distribution as a multinomial facilitates conjugacy in the model and enables simple and efficient Gibbs sampling and variational Bayes (VB) inference updates, with a computational cost that only depends on the number of nonzeros in the tensor. The model also provides a nice interpretability for the factors; in our model, each factor corresponds to a "topic". We develop a set of online inference algorithms that allow further scaling up the model to massive tensors, for which batch inference methods may be infeasible. We apply our framework on diverse real-world applications, such as \emph{multiway} topic modeling on a scientific publications database, analyzing a political science data set, and analyzing a massive household transactions data set.

We present a scalable Bayesian model for low-rank factorization of massive tensors with binary observations. The proposed model has the following key properties: (1) in contrast to the models based on the logistic or probit likelihood, using a zero-truncated Poisson likelihood for binary data allows our model to scale up in the number of \emph{ones} in the tensor, which is especially appealing for massive but sparse binary tensors; (2) side-information in form of binary pairwise relationships (e.g., an adjacency network) between objects in any tensor mode can also be leveraged, which can be especially useful in "cold-start" settings; and (3) the model admits simple Bayesian inference via batch, as well as \emph{online} MCMC; the latter allows scaling up even for \emph{dense} binary data (i.e., when the number of ones in the tensor/network is also massive). In addition, non-negative factor matrices in our model provide easy interpretability, and the tensor rank can be inferred from the data. We evaluate our model on several large-scale real-world binary tensors, achieving excellent computational scalability, and also demonstrate its usefulness in leveraging side-information provided in form of mode-network(s).

Recursive neural models, which use syntactic parse trees to recursively generate representations bottom-up, are a popular architecture. But there have not been rigorous evaluations showing for exactly which tasks this syntax-based method is appropriate. In this paper we benchmark {\bf recursive} neural models against sequential {\bf recurrent} neural models (simple recurrent and LSTM models), enforcing apples-to-apples comparison as much as possible. We investigate 4 tasks: (1) sentiment classification at the sentence level and phrase level; (2) matching questions to answer-phrases; (3) discourse parsing; (4) semantic relation extraction (e.g., {\em component-whole} between nouns). Our goal is to understand better when, and why, recursive models can outperform simpler models. We find that recursive models help mainly on tasks (like semantic relation extraction) that require associating headwords across a long distance, particularly on very long sequences. We then introduce a method for allowing recurrent models to achieve similar performance: breaking long sentences into clause-like units at punctuation and processing them separately before combining. Our results thus help understand the limitations of both classes of models, and suggest directions for improving recurrent models.

We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.

In this paper I present a new approach for regression of time series using their own samples. This is a celebrated problem known as Auto-Regression. Dealing with outlier or missed samples in a time series makes the problem of estimation difficult, so it should be robust against them. Moreover for coding purposes I will show that it is desired the residual of auto-regression be sparse. To these aims, I first assume a multivariate Gaussian prior on the residual and then obtain the estimation. Two simple simulations have been done on spectrum estimation and speech coding.

The interaction of multiple autonomous agents gives rise to highly dynamic and nondeterministic environments, contributing to the complexity in applications such as automated financial markets, smart grids, or robotics. Due to the sheer number of situations that may arise, it is not possible to foresee and program the optimal behaviour for all agents beforehand. Consequently, it becomes essential for the success of the system that the agents can learn their optimal behaviour and adapt to new situations or circumstances. The past two decades have seen the emergence of reinforcement learning, both in single and multi-agent settings, as a strong, robust and adaptive learning paradigm. Progress has been substantial, and a wide range of algorithms are now available. An important challenge in the domain of multi-agent learning is to gain qualitative insights into the resulting system dynamics. In the past decade, tools and methods from evolutionary game theory have been successfully employed to study multi-agent learning dynamics formally in strategic interactions. This article surveys the dynamical models that have been derived for various multi-agent reinforcement learning algorithms, making it possible to study and compare them qualitatively. Furthermore, new learning algorithms that have been introduced using these evolutionary game theoretic tools are reviewed. The evolutionary models can be used to study complex strategic interactions. Examples of such analysis are given for the domains of automated trading in stock markets and collision avoidance in multi-robot systems. The paper provides a roadmap on the progress that has been achieved in analysing the evolutionary dynamics of multi-agent learning by highlighting the main results and accomplishments.

In the era of deep learning several unsupervised models have been developed to capture the key features in unlabeled handwritten data. Popular among them is the Restricted Boltzmann Machines RBM. However, due to the novelty in handwritten multidialect data, the RBM may fail to generate an efficient representation. In this paper we propose a generative model, the Mode Synthesizing Machine MSM for on-line representation of real life handwritten multidialect language data. The MSM takes advantage of the hierarchical representation of the modes of a data distribution using a two-point error update to learn a sequence of representative multidialects in a generative way. Experiments were performed to evaluate the performance of the MSM over the RBM with the former attaining much lower error values than the latter on both independent and mixed data set.