Asymptotic Normality of the Maximum Pseudolikelihood Estimator for Fully Visible Boltzmann Machines

Boltzmann machines (BMs) are a class of binary neural networks for which there have been numerous proposed methods of estimation. Recently, it has been shown that in the fully visible case of the BM, the method of maximum pseudolikelihood estimation (MPLE) results in parameter estimates which are consistent in the probabilistic sense. In this article, we investigate the properties of MPLE for the fully visible BMs further, and prove that MPLE also yields an asymptotically normal parameter estimator. These results can be used to construct confidence intervals and to test statistical hypotheses. We support our theoretical results by showing that the estimator behaves as expected in a simulation study

A Deep Embedding Model for Co-occurrence Learning

Co-occurrence Data is a common and important information source in many areas, such as the word co-occurrence in the sentences, friends co-occurrence in social networks and products co-occurrence in commercial transaction data, etc, which contains rich correlation and clustering information about the items. In this paper, we study co-occurrence data using a general energy-based probabilistic model, and we analyze three different categories of energy-based model, namely, the $L_1$, $L_2$ and $L_k$ models, which are able to capture different levels of dependency in the co-occurrence data. We also discuss how several typical existing models are related to these three types of energy models, including the Fully Visible Boltzmann Machine (FVBM) ($L_2$), Matrix Factorization ($L_2$), Log-BiLinear (LBL) models ($L_2$), and the Restricted Boltzmann Machine (RBM) model ($L_k$). Then, we propose a Deep Embedding Model (DEM) (an $L_k$ model) from the energy model in a \emph{principled} manner. Furthermore, motivated by the observation that the partition function in the energy model is intractable and the fact that the major objective of modeling the co-occurrence data is to predict using the conditional probability, we apply the \emph{maximum pseudo-likelihood} method to learn DEM. In consequence, the developed model and its learning method naturally avoid the above difficulties and can be easily used to compute the conditional probability in prediction. Interestingly, our method is equivalent to learning a special structured deep neural network using back-propagation and a special sampling strategy, which makes it scalable on large-scale datasets. Finally, in the experiments, we show that the DEM can achieve comparable or better results than state-of-the-art methods on datasets across several application domains

Unsupervised Generative Modeling Using Matrix Product States

Generative modeling, which learns joint probability distribution from data and generates samples according to it, is an important task in machine learning and artificial intelligence. Inspired by probabilistic interpretation of quantum physics, we propose a generative model using matrix product states, which is a tensor network originally proposed for describing (particularly one-dimensional) entangled quantum states. Our model enjoys efficient learning analogous to the density matrix renormalization group method, which allows dynamically adjusting dimensions of the tensors and offers an efficient direct sampling approach for generative tasks. We apply our method to generative modeling of several standard datasets including the Bars and Stripes, random binary patterns and the MNIST handwritten digits to illustrate the abilities, features and drawbacks of our model over popular generative models such as Hopfield model, Boltzmann machines and generative adversarial networks. Our work sheds light on many interesting directions of future exploration on the development of quantum-inspired algorithms for unsupervised machine learning, which are promisingly possible to be realized on quantum devices.Comment: 11 pages, 12 figures (not including the TNs) GitHub Page: https://congzlwag.github.io/UnsupGenModbyMPS

Fast maximum likelihood estimation via equilibrium expectation for large network data

This is the final version. Available from the publisher via the DOI in this record.A major line of contemporary research on complex networks is based on the development of statistical models that specify the local motifs associated with macro-structural properties observed in actual networks. This statistical approach becomes increasingly problematic as network size increases. In the context of current research on efficient estimation of models for large network data sets, we propose a fast algorithm for maximum likelihood estimation (MLE) that affords a significant increase in the size of networks amenable to direct empirical analysis. The algorithm we propose in this paper relies on properties of Markov chains at equilibrium, and for this reason it is called equilibrium expectation (EE). We demonstrate the performance of the EE algorithm in the context of exponential random graph models (ERGMs) a family of statistical models commonly used in empirical research based on network data observed at a single period in time. Thus far, the lack of efficient computational strategies has limited the empirical scope of ERGMs to relatively small networks with a few thousand nodes. The approach we propose allows a dramatic increase in the size of networks that may be analyzed using ERGMs. This is illustrated in an analysis of several biological networks and one social network with 104,103 nodes.Swiss National Science Foundatio