In this study, we consider an empirical Bayes method for Boltzmann machines and propose an algorithm for it. The empirical Bayes method allows estimation of the values of the hyperparameters of the Boltzmann machine by maximizing a specific likelihood function referred to as the empirical Bayes likelihood function in this study. However, the maximization is computationally hard because the empirical Bayes likelihood function involves intractable integrations of the partition function. The proposed algorithm avoids this computational problem by using the replica method and the Plefka expansion. Our method does not require any iterative procedures and is quite simple and fast, though it introduces a bias to the estimate, which exhibits an unnatural behavior with respect to the size of the dataset. This peculiar behavior is supposed to be due to the approximate treatment by the Plefka expansion. A possible extension to overcome this behavior is also discussed.

While likelihood-based inference and its variants provide a statistically efficient and widely applicable approach to parametric inference, their application to models involving intractable likelihoods poses challenges. In this work, we study a class of minimum distance estimators for intractable generative models, that is, statistical models for which the likelihood is intractable, but simulation is cheap. The distance considered, maximum mean discrepancy (MMD), is defined through the embedding of probability measures into a reproducing kernel Hilbert space. We study the theoretical properties of these estimators, showing that they are consistent, asymptotically normal and robust to model misspecification. A main advantage of these estimators is the flexibility offered by the choice of kernel, which can be used to trade-off statistical efficiency and robustness. On the algorithmic side, we study the geometry induced by MMD on the parameter space and use this to introduce a novel natural gradient descent-like algorithm for efficient implementation of these estimators. We illustrate the relevance of our theoretical results on several classes of models including a discrete-time latent Markov process and two multivariate stochastic differential equation models.

We introduce and demonstrate the variational autoencoder (VAE) for probabilistic non-negative matrix factorisation (PAE-NMF). We design a network which can perform non-negative matrix factorisation (NMF) and add in aspects of a VAE to make the coefficients of the latent space probabilistic. By restricting the weights in the final layer of the network to be non-negative and using the non-negative Weibull distribution we produce a probabilistic form of NMF which allows us to generate new data and find a probability distribution that effectively links the latent and input variables. We demonstrate the effectiveness of PAE-NMF on three heterogeneous datasets: images, financial time series and genomic.

We consider the online linear regression problem, where the predictor vector may vary with time. This problem can be modelled as a linear dynamical system, where the parameters that need to be learned are the variance of both the process noise and the observation noise. The classical approach to learning the variance is via the maximum likelihood estimator -- a non-convex optimization problem prone to local minima and with no finite sample complexity bounds. In this paper we study the global system operator: the operator that maps the noises vectors to the output. In particular, we obtain estimates on its spectrum, and as a result derive the first known variance estimators with sample complexity guarantees for online regression problems. We demonstrate the approach on a number of synthetic and real-world benchmarks.

The calibration of a reservoir model with observed transient data of fluid pressures and rates is a key task in obtaining a predictive model of the flow and transport behaviour of the earth's subsurface. The model calibration task, commonly referred to as "history matching", can be formalised as an ill-posed inverse problem where we aim to find the underlying spatial distribution of petrophysical properties that explain the observed dynamic data. We use a generative adversarial network pretrained on geostatistical object-based models to represent the distribution of rock properties for a synthetic model of a hydrocarbon reservoir. The dynamic behaviour of the reservoir fluids is modelled using a transient two-phase incompressible Darcy formulation. We invert for the underlying reservoir properties by first modeling property distributions using the pre-trained generative model then using the adjoint equations of the forward problem to perform gradient descent on the latent variables that control the output of the generative model. In addition to the dynamic observation data, we include well rock-type constraints by introducing an additional objective function. Our contribution shows that for a synthetic test case, we are able to obtain solutions to the inverse problem by optimising in the latent variable space of a deep generative model, given a set of transient observations of a non-linear forward problem.

Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist. We provide a precise analysis of the performance of RLR via the solution of a system of six nonlinear equations, through which any performance metric of interest (mean, mean-squared error, probability of support recovery, etc.) can be explicitly computed. Our results generalize those of Sur and Candes and we provide a detailed study for the cases of $\ell_2^2$-RLR and sparse ($\ell_1$-regularized) logistic regression. In both cases, we obtain explicit expressions for various performance metrics and can find the values of the regularizer parameter that optimizes the desired performance. The theory is validated by extensive numerical simulations across a range of parameter values and problem instances.

Variational inference for Bayesian deep neural networks (DNNs) requires specifying priors and approximate posterior distributions for neural network weights. Specifying meaningful weight priors is a challenging problem, particularly for scaling variational inference to deeper architectures involving high dimensional weight space. We propose Bayesian MOdel Priors Extracted from Deterministic DNN (MOPED) method for stochastic variational inference to choose meaningful prior distributions over weight space using deterministic weights derived from the pretrained DNNs of equivalent architecture. We evaluate the proposed approach on multiple datasets and real-world application domains with a range of varying complex model architectures to demonstrate MOPED enables scalable variational inference for Bayesian DNNs. The proposed method achieves faster training convergence and provides reliable uncertainty quantification, without compromising on the accuracy provided by the deterministic DNNs. We also propose hybrid architectures to Bayesian DNNs where deterministic and variational layers are combined to balance computation complexity during prediction phase and while providing benefits of Bayesian inference. We will release the source code for this work.

Collaborative representation is a popular feature learning approach, which encoding process is assisted by variety types of information. In this paper, we propose a collaborative representation restricted Boltzmann Machine (CRRBM) for modeling binary data and a collaborative representation Gaussian restricted Boltzmann Machine (CRGRBM) for modeling realvalued data by applying a collaborative representation strategy in the encoding procedure. We utilize Locality Sensitive Hashing (LSH) to generate similar sample subsets of the instance and observed feature set simultaneously from input data. Hence, we can obtain some mini blocks, which come from the intersection of instance and observed feature subsets. Then we integrate Contrastive Divergence and Bregman Divergence methods with mini blocks to optimize our CRRBM and CRGRBM models. In their training process, the complex collaborative relationships between multiple instances and features are fused into the hidden layer encoding. Hence, these encodings have dual characteristics of concealment and cooperation. Here, we develop two deep collaborative encoder frameworks (DCEF) based on the CRRBM and CRGRBM models: one is a DCEF with Gaussian linear visible units (GDCEF) for modeling real-valued data, and the other is a DCEF with binary visible units (BDCEF) for modeling binary data. We explore the collaborative representation capability of the hidden features in every layer of the GDCEF and BDCEF framework, especially in the deepest hidden layer. The experimental results show that the GDCEF and BDCEF frameworks have more outstanding performances than the classic Autoencoder framework for unsupervised clustering task on the MSRA-MM2.0 and UCI datasets, respectively.

Many applications for classification methods not only require high accuracy but also reliable estimation of predictive uncertainty. However, while many current classification frameworks, in particular deep neural network architectures, provide very good results in terms of accuracy, they tend to underestimate their predictive uncertainty. In this paper, we propose a method that corrects the confidence output of a general classifier such that it approaches the true probability of classifying correctly. This classifier calibration is, in contrast to existing approaches, based on a non-parametric representation using a latent Gaussian process and specifically designed for multi-class classification. It can be applied to any classification method that outputs confidence estimates and is not limited to neural networks. We also provide a theoretical analysis regarding the over- and underconfidence of a classifier and its relationship to calibration. In experiments we show the universally strong performance of our method across different classifiers and benchmark data sets in contrast to existing classifier calibration techniques.

This thesis presents new methods for unsupervised learning of distributed representations of words and entities from text and knowledge bases. The first algorithm presented in the thesis is a multi-view algorithm for learning representations of words called Multiview Latent Semantic Analysis (MVLSA). By incorporating up to 46 different types of co-occurrence statistics for the same vocabulary of english words, I show that MVLSA outperforms other state-of-the-art word embedding models. Next, I focus on learning entity representations for search and recommendation and present the second method of this thesis, Neural Variational Set Expansion (NVSE). NVSE is also an unsupervised learning method, but it is based on the Variational Autoencoder framework. Evaluations with human annotators show that NVSE can facilitate better search and recommendation of information gathered from noisy, automatic annotation of unstructured natural language corpora. Finally, I move from unstructured data and focus on structured knowledge graphs. I present novel approaches for learning embeddings of vertices and edges in a knowledge graph that obey logical constraints.