Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.

We propose a method for automatically answering questions about images by bringing together recent advances from natural language processing and computer vision. We combine discrete reasoning with uncertain predictions by a multi-world approach that represents uncertainty about the perceived world in a bayesian framework. Our approach can handle human questions of high complexity about realistic scenes and replies with range of answer like counts, object classes, instances and lists of them. The system is directly trained from question-answer pairs. We establish a first benchmark for this task that can be seen as a modern attempt at a visual turing test.

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

Selecting the right algorithm is an important problem in computer science, because the algorithm often has to exploit the structure of the input to be efficient. The human mind faces the same challenge. Therefore, solutions to the algorithm selection problem can inspire models of human strategy selection and vice versa. Here, we view the algorithm selection problem as a special case of metareasoning and derive a solution that outperforms existing methods in sorting algorithm selection. We apply our theory to model how people choose between cognitive strategies and test its prediction in a behavioral experiment. We find that people quickly learn to adaptively choose between cognitive strategies. People's choices in our experiment are consistent with our model but inconsistent with previous theories of human strategy selection. Rational metareasoning appears to be a promising framework for reverse-engineering how people choose among cognitive strategies and translating the results into better solutions to the algorithm selection problem.

We present a method for estimating articulated human pose from a single static image based on a graphical model with novel pairwise relations that make adaptive use of local image measurements. More precisely, we specify a graphical model for human pose which exploits the fact the local image measurements can be used both to detect parts (or joints) and also to predict the spatial relationships between them (Image Dependent Pairwise Relations). These spatial relationships are represented by a mixture model. We use Deep Convolutional Neural Networks (DCNNs) to learn conditional probabilities for the presence of parts and their spatial relationships within image patches. Hence our model combines the representational flexibility of graphical models with the efficiency and statistical power of DCNNs. Our method significantly outperforms the state of the art methods on the LSP and FLIC datasets and also performs very well on the Buffy dataset without any training.

Dropout training, originally designed for deep neural networks, has been successful on high-dimensional single-layer natural language tasks. This paper proposes a theoretical explanation for this phenomenon: we show that, under a generative Poisson topic model with long documents, dropout training improves the exponent in the generalization bound for empirical risk minimization. Dropout achieves this gain much like a marathon runner who practices at altitude: once a classifier learns to perform reasonably well on training examples that have been artificially corrupted by dropout, it will do very well on the uncorrupted test set. We also show that, under similar conditions, dropout preserves the Bayes decision boundary and should therefore induce minimal bias in high dimensions.

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers.

We extend generative stochastic networks to supervised learning of representations. In particular, we introduce a hybrid training objective considering a generative and discriminative cost function governed by a trade-off parameter lambda. We use a new variant of network training involving noise injection, i.e. walkback training, to jointly optimize multiple network layers. Neither additional regularization constraints, such as l1, l2 norms or dropout variants, nor pooling- or convolutional layers were added. Nevertheless, we are able to obtain state-of-the-art performance on the MNIST dataset, without using permutation invariant digits and outperform baseline models on sub-variants of the MNIST and rectangles dataset significantly.

Neural population activity in cortical circuits is not solely driven by external inputs, but is also modulated by endogenous states which vary on multiple time-scales. To understand information processing in cortical circuits, we need to understand the statistical structure of internal states and their interaction with sensory inputs. Here, we present a statistical model for extracting hierarchically organised neural population states from multi-channel recordings of neural spiking activity. Population states are modelled using a hidden Markov decision tree with state-dependent tuning parameters and a generalised linear observation model. We present a variational Bayesian inference algorithm for estimating the posterior distribution over parameters from neural population recordings. On simulated data, we show that we can identify the underlying sequence of population states and reconstruct the ground truth parameters. Using population recordings from visual cortex, we find that a model with two levels of population states outperforms both a one-state and a two-state generalised linear model. Finally, we find that modelling of state-dependence also improves the accuracy with which sensory stimuli can be decoded from the population response.

Performance metrics for binary classification are designed to capture tradeoffs between four fundamental population quantities: true positives, false positives, true negatives and false negatives. Despite significant interest from theoretical and applied communities, little is known about either optimal classifiers or consistent algorithms for optimizing binary classification performance metrics beyond a few special cases. We consider a fairly large family of performance metrics given by ratios of linear combinations of the four fundamental population quantities. This family includes many well known binary classification metrics such as classification accuracy, AM measure, F-measure and the Jaccard similarity coefficient as special cases. Our analysis identifies the optimal classifiers as the sign of the thresholded conditional probability of the positive class, with a performance metric-dependent threshold. The optimal threshold can be constructed using simple plug-in estimators when the performance metric is a linear combination of the population quantities, but alternative techniques are required for the general case. We propose two algorithms for estimating the optimal classifiers, and prove their statistical consistency. Both algorithms are straightforward modifications of standard approaches to address the key challenge of optimal threshold selection, thus are simple to implement in practice. The first algorithm combines a plug-in estimate of the conditional probability of the positive class with optimal threshold selection. The second algorithm leverages recent work on calibrated asymmetric surrogate losses to construct candidate classifiers. We present empirical comparisons between these algorithms on benchmark datasets.