We describe a latent variable model for supervised dimensionality reduction and distance metric learning. The model discovers linear projections of high dimensional data that shrink the distance between similarly labeled inputs and expand the distance between differently labeled ones. The model’s continuous latent variables locate pairs of examples in a latent space of lower dimensionality. The model differs significantly from classical factor analysis in that the posterior distribution over these latent variables is not always multivariate Gaussian. Nevertheless we show that inference is completely tractable and derive an Expectation-Maximization (EM) algorithm for parameter estimation. We also compare the model to other approaches in distance metric learning. The model’s main advantage is its simplicity: at each iteration of the EM algorithm, the distance metric is re-estimated by solving an unconstrained least-squares problem. Experiments show that these simple updates are highly effective.

We introduce a large-volume box classification for binary prediction, which maintains a subset of weight vectors, and specifically axis-aligned boxes. Our learning algorithm seeks for a box of large volume that contains ``simple'' weight vectors which most of are accurate on the training set. Two versions of the learning process are cast as convex optimization problems, and it is shown how to solve them efficiently. The formulation yields a natural PAC-Bayesian performance bound and it is shown to minimize a quantity directly aligned with it. The algorithm outperforms SVM and the recently proposed AROW algorithm on a majority of $30$ NLP datasets and binarized USPS optical character recognition datasets.

A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common structure underlyingrelations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a flexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efficient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases.

For modeling data matrices, this paper introduces Probabilistic Co-Subspace Addition (PCSA) model by simultaneously capturing the dependent structures among both rows and columns. Briefly, PCSA assumes that each entry of a matrix is generated by the additive combination of the linear mappings of two features, which distribute in the row-wise and column-wise latent subspaces. Consequently, it captures the dependencies among entries intricately, and is able to model the non-Gaussian and heteroscedastic density. Variational inference is proposed on PCSA for approximate Bayesian learning, where the updating for posteriors is formulated into the problem of solving Sylvester equations. Furthermore, PCSA is extended to tackling and filling missing values, to adapting its sparseness, and to modelling tensor data. In comparison with several state-of-art approaches, experiments demonstrate the effectiveness and efficiency of Bayesian (sparse) PCSA on modeling matrix (tensor) data and filling missing values.

Recent spiking network models of Bayesian inference and unsupervised learning frequently assume either inputs to arrive in a special format or employ complex computations in neuronal activation functions and synaptic plasticity rules. Here we show in a rigorous mathematical treatment how homeostatic processes, which have previously received little attention in this context, can overcome common theoretical limitations and facilitate the neural implementation and performance of existing models. In particular, we show that homeostatic plasticity can be understood as the enforcement of a 'balancing' posterior constraint during probabilistic inference and learning with Expectation Maximization. We link homeostatic dynamics to the theory of variational inference, and show that nontrivial terms, which typically appear during probabilistic inference in a large class of models, drop out. We demonstrate the feasibility of our approach in a spiking Winner-Take-All architecture of Bayesian inference and learning. Finally, we sketch how the mathematical framework can be extended to richer recurrent network architectures. Altogether, our theory provides a novel perspective on the interplay of homeostatic processes and synaptic plasticity in cortical microcircuits, and points to an essential role of homeostasis during inference and learning in spiking networks.

Continuous relaxations play an important role in discrete optimization, but have not seen much use in approximate probabilistic inference. Here we show that a general form of the Gaussian Integral Trick makes it possible to transform a wide class of discrete variable undirected models into fully continuous systems. The continuous representation allows the use of gradient-based Hamiltonian Monte Carlo for inference, results in new ways of estimating normalization constants (partition functions), and in general opens up a number of new avenues for inference in difficult discrete systems. We demonstrate some of these continuous relaxation inference algorithms on a number of illustrative problems.

Links between probabilistic and non-probabilistic learning algorithms can arise by performing small-variance asymptotics, i.e., letting the variance of particular distributions in a graphical model go to zero. For instance, in the context of clustering, such an approach yields precise connections between the k-means and EM algorithms. In this paper, we explore small-variance asymptotics for exponential family Dirichlet process (DP) and hierarchical Dirichlet process (HDP) mixture models. Utilizing connections between exponential family distributions and Bregman divergences, we derive novel clustering algorithms from the asymptotic limit of the DP and HDP mixtures that feature the scalability of existing hard clustering methods as well as the flexibility of Bayesian nonparametric models. We focus on special cases of our analysis for discrete-data problems, including topic modeling, and we demonstrate the utility of our results by applying variants of our algorithms to problems arising in vision and document analysis.

We present a new variational inference algorithm for Gaussian process regression withnon-conjugate likelihood functions, with application to a wide array of problems including binary and multi-class classification, and ordinal regression. Our method constructs a concave lower bound that is optimized using an efficient fixed-point updating algorithm. We show that the new algorithm has highly competitive computationalcomplexity, matching that of alternative approximate inference methods. We also prove that the use of concave variational bounds provides stable and guaranteed convergence - a property not available to other approaches. We show empirically for both binary and multi-class classification that our new algorithm converges much faster than existing variational methods, and without any degradation in performance.

Accurate and detailed models of the progression of neurodegenerative diseases such as Alzheimer's (AD) are crucially important for reliable early diagnosis and the determination and deployment of effective treatments. In this paper, we introduce the ALPACA (Alzheimer's disease Probabilistic Cascades) model, a generative model linking latent Alzheimer's progression dynamics to observable biomarker data. In contrast with previous works which model disease progression as a fixed ordering of events, we explicitly model the variability over such orderings among patients which is more realistic, particularly for highly detailed disease progression models. We describe efficient learning algorithms for ALPACA and discuss promising experimental results on a real cohort of Alzheimer's patients from the Alzheimer's Disease Neuroimaging Initiative.

Recent experiments have demonstrated that humans and animals typically reason probabilistically about their environment. This ability requires a neural code that represents probability distributions and neural circuits that are capable of implementing the operations of probabilistic inference. The proposed probabilistic population coding (PPC) framework provides a statistically efficient neural representation of probability distributions that is both broadly consistent with physiological measurements and capable of implementing some of the basic operations of probabilistic inference in a biologically plausible way. However, these experiments and the corresponding neural models have largely focused on simple (tractable) probabilistic computations such as cue combination, coordinate transformations, and decision making. As a result it remains unclear how to generalize this framework to more complex probabilistic computations. Here we address this short coming by showing that a very general approximate inference algorithm known as Variational Bayesian Expectation Maximization can be implemented within the linear PPC framework. We apply this approach to a generic problem faced by any given layer of cortex, namely the identification of latent causes of complex mixtures of spikes. We identify a formal equivalent between this spike pattern demixing problem and topic models used for document classification, in particular Latent Dirichlet Allocation (LDA). We then construct a neural network implementation of variational inference and learning for LDA that utilizes a linear PPC. This network relies critically on two non-linear operations: divisive normalization and super-linear facilitation, both of which are ubiquitously observed in neural circuits. We also demonstrate how online learning can be achieved using a variation of Hebb’s rule and describe an extesion of this work which allows us to deal with time varying and correlated latent causes.