We study optimal image encoding based on a generative approach with non-linear feature combinations and explicit position encoding. By far most approaches to unsupervised learning learning of visual features, such as sparse coding or ICA, account for translations by representing the same features at different positions. Some earlier models used a separate encoding of features and their positions to facilitate invariant data encoding and recognition. All probabilistic generative models with explicit position encoding have so far assumed a linear superposition of components to encode image patches. Here, we for the first time apply a model with non-linear feature superposition and explicit position encoding. By avoiding linear superpositions, the studied model represents a closer match to component occlusions which are ubiquitous in natural images. In order to account for occlusions, the non-linear model encodes patches qualitatively very different from linear models by using component representations separated into mask and feature parameters. We first investigated encodings learned by the model using artificial data with mutually occluding components. We find that the model extracts the components, and that it can correctly identify the occlusive components with the hidden variables of the model. On natural image patches, the model learns component masks and features for typical image components. By using reverse correlation, we estimate the receptive fields associated with the model's hidden units. We find many Gabor-like or globular receptive fields as well as fields sensitive to more complex structures. Our results show that probabilistic models that capture occlusions and invariances can be trained efficiently on image patches, and that the resulting encoding represents an alternative model for the neural encoding of images in the primary visual cortex.

Infinite mixture models are commonly used for clustering. One can sample from the posterior of mixture assignments by Monte Carlo methods or find its maximum a posteriori solution by optimization. However, in some problems the posterior is diffuse and it is hard to interpret the sampled partitionings. In this paper, we introduce novel statistics based on block sizes for representing sample sets of partitionings and feature allocations. We develop an element-based definition of entropy to quantify segmentation among their elements. Then we propose a simple algorithm called entropy agglomeration (EA) to summarize and visualize this information. Experiments on various infinite mixture posteriors as well as a feature allocation dataset demonstrate that the proposed statistics are useful in practice.

Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parametrization of joint distributions. Among other properties, copulas provide a recipe for combining flexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. Inference is typically done in a Bayesian framework with Gaussian copulas, and it is complicated by the fact this implies sampling within a space where the number of constraints increase quadratically with the number of data points. The result is slow mixing when using off-the-shelf Gibbs sampling. We present an efficient algorithm based on recent advances on constrained Hamiltonian Markov chain Monte Carlo that is simple to implement and does not require paying for a quadratic cost in sample size.

We present SDA-Bayes, a framework for (S)treaming, (D)istributed, (A)synchronous computation of a Bayesian posterior. The framework makes streaming updates to the estimated posterior according to a user-specified approximation primitive function. We demonstrate the usefulness of our framework, with variational Bayes (VB) as the primitive, by fitting the latent Dirichlet allocation model to two large-scale document collections. We demonstrate the advantages of our algorithm over stochastic variational inference (SVI), both in the single-pass setting SVI was designed for and in the streaming setting, to which SVI does not apply.

Monte-Carlo tree search is drawing great interest in the domain of planning under uncertainty, particularly when little or no domain knowledge is available. One of the central problems is the trade-off between exploration and exploitation. In this paper we present a novel Bayesian mixture modelling and inference based Thompson sampling approach to addressing this dilemma. The proposed Dirichlet-NormalGamma MCTS (DNG-MCTS) algorithm represents the uncertainty of the accumulated reward for actions in the MCTS search tree as a mixture of Normal distributions and inferences on it in Bayesian settings by choosing conjugate priors in the form of combinations of Dirichlet and NormalGamma distributions. Thompson sampling is used to select the best action at each decision node. Experimental results show that our proposed algorithm has achieved the state-of-the-art comparing with popular UCT algorithm in the context of online planning for general Markov decision processes.

Variational inference algorithms provide the most effective framework for large-scale training of Bayesian nonparametric models. Stochastic online approaches are promising, but are sensitive to the chosen learning rate and often converge to poor local optima. We present a new algorithm, memoized online variational inference, which scales to very large (yet finite) datasets while avoiding the complexities of stochastic gradient. Our algorithm maintains finite-dimensional sufficient statistics from batches of the full dataset, requiring some additional memory but still scaling to millions of examples. Exploiting nested families of variational bounds for infinite nonparametric models, we develop principled birth and merge moves allowing non-local optimization. Births adaptively add components to the model to escape local optima, while merges remove redundancy and improve speed. Using Dirichlet process mixture models for image clustering and denoising, we demonstrate major improvements in robustness and accuracy.

Learning a visual concept from a small number of positive examples is a significant challenge for machine learning algorithms. Current methods typically fail to find the appropriate level of generalization in a concept hierarchy for a given set of visual examples. Recent work in cognitive science on Bayesian models of generalization addresses this challenge, but prior results assumed that objects were perfectly recognized. We present an algorithm for learning visual concepts directly from images, using probabilistic predictions generated by visual classifiers as the input to a Bayesian generalization model. As no existing challenge data tests this paradigm, we collect and make available a new, large-scale dataset for visual concept learning using the ImageNet hierarchy as the source of possible concepts, with human annotators to provide ground truth labels as to whether a new image is an instance of each concept using a paradigm similar to that used in experiments studying word learning in children. We compare the performance of our system to several baseline algorithms, and show a significant advantage results from combining visual classifiers with the ability to identify an appropriate level of abstraction using Bayesian generalization.

Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when computing conditional probabilities. The reason is that conditioning on evidence breaks many of the model's symmetries, which preempts standard lifting techniques. Recent theoretical results show, for example, that conditioning on evidence which corresponds to binary relations is #P-hard, suggesting that no lifting is to be expected in the worst case. In this paper, we balance this grim result by identifying the Boolean rank of the evidence as a key parameter for characterizing the complexity of conditioning in lifted inference. In particular, we show that conditioning on binary evidence with bounded Boolean rank is efficient. This opens up the possibility of approximating evidence by a low-rank Boolean matrix factorization, which we investigate both theoretically and empirically.

We consider a number of classical and new computational problems regarding marginal distributions, and inference in models specifying a full joint distribution. We prove general and efficient reductions between a number of these problems, which demonstrate that algorithmic progress in inference automatically yields progress for "pure data" problems. Our main technique involves formulating the problems as linear programs, and proving that the dual separation oracle required by the ellipsoid method is provided by the target problem. This technique may be of independent interest in probabilistic inference.

What if there is a teacher who knows the learning goal and wants to design good training data for a machine learner? We propose an optimal teaching framework aimed at learners who employ Bayesian models. Our framework is expressed as an optimization problem over teaching examples that balance the future loss of the learner and the effort of the teacher. This optimization problem is in general hard. In the case where the learner employs conjugate exponential family models, we present an approximate algorithm for finding the optimal teaching set. Our algorithm optimizes the aggregate sufficient statistics, then unpacks them into actual teaching examples. We give several examples to illustrate our framework.