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.

Neural responses in visual cortex are influenced by visual stimuli and by ongoing spiking activity in local circuits. An important challenge in computational neuroscience is to develop models that can account for both of these features in large multi-neuron recordings and to reveal how stimulus representations interact with and depend on cortical dynamics. Here we introduce a statistical model of neural population activity that integrates a nonlinear receptive field model with a latent dynamical model of ongoing cortical activity. This model captures the temporal dynamics, effective network connectivity in large population recordings, and correlations due to shared stimulus drive as well as common noise. Moreover, because the nonlinear stimulus inputs are mixed by the ongoing dynamics, the model can account for a relatively large number of idiosyncratic receptive field shapes with a small number of nonlinear inputs to a low-dimensional latent dynamical model. We introduce a fast estimation method using online expectation maximization with Laplace approximations. Inference scales linearly in both population size and recording duration. We apply this model to multi-channel recordings from primary visual cortex and show that it accounts for a large number of individual neural receptive fields using a small number of nonlinear inputs and a low-dimensional dynamical model.

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.

The brain uses population codes to form distributed, noise-tolerant representations of sensory and motor variables. Recent work has examined the theoretical optimality of such codes in order to gain insight into the principles governing population codes found in the brain. However, the majority of the population coding literature considers either conditionally independent neurons or neurons with noise governed by a stimulus-independent covariance matrix. Here we analyze population coding under a simple alternative model in which latent input noise" corrupts the stimulus before it is encoded by the population. This provides a convenient and tractable description for irreducible uncertainty that cannot be overcome by adding neurons, and induces stimulus-dependent correlations that mimic certain aspects of the correlations observed in real populations. We examine prior-dependent, Bayesian optimal coding in such populations using exact analyses of cases in which the posterior is approximately Gaussian. These analyses extend previous results on independent Poisson population codes and yield an analytic expression for squared loss and a tight upper bound for mutual information. We show that, for homogeneous populations that tile the input domain, optimal tuning curve width depends on the prior, the loss function, the resource constraint, and the amount of input noise. This framework provides a practical testbed for examining issues of optimality, noise, correlation, and coding fidelity in realistic neural populations."

This paper proposes a new hybrid architecture that consists of a deep Convolutional Network and a Markov Random Field. We show how this architecture is successfully applied to the challenging problem of articulated human pose estimation in monocular images. The architecture can exploit structural domain constraints such as geometric relationships between body joint locations. We show that joint training of these two model paradigms improves performance and allows us to significantly outperform existing state-of-the-art techniques.

We study online combinatorial decision problems, where one must make sequential decisions in some combinatorial space without knowing in advance the cost of decisions on each trial; the goal is to minimize the total regret over some sequence of trials relative to the best fixed decision in hindsight. Such problems have been studied mostly in settings where decisions are represented by Boolean vectors and costs are linear in this representation. Here we study a general setting where costs may be linear in any suitable low-dimensional vector representation of elements of the decision space. We give a general algorithm for such problems that we call low-dimensional online mirror descent (LDOMD); the algorithm generalizes both the Component Hedge algorithm of Koolen et al. (2010), and a recent algorithm of Suehiro et al. (2012). Our study offers a unification and generalization of previous work, and emphasizes the role of the convex polytope arising from the vector representation of the decision space; while Boolean representations lead to 0-1 polytopes, more general vector representations lead to more general polytopes. We study several examples of both types of polytopes. Finally, we demonstrate the benefit of having a general framework for such problems via an application to an online transportation problem; the associated transportation polytopes generalize the Birkhoff polytope of doubly stochastic matrices, and the resulting algorithm generalizes the PermELearn algorithm of Helmbold and Warmuth (2009).

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.

In this paper we propose a framework for supervised and semi-supervised learning based on reformulating the learning problem as a regularized Fredholm integral equation. Our approach fits naturally into the kernel framework and can be interpreted as constructing new data-dependent kernels, which we call Fredholm kernels. We proceed to discuss the noise assumption" for semi-supervised learning and provide evidence evidence both theoretical and experimental that Fredholm kernels can effectively utilize unlabeled data under the noise assumption. We demonstrate that methods based on Fredholm learning show very competitive performance in the standard semi-supervised learning setting."

In many situations we have some measurement of confidence on ``positiveness for a binary label. The ``positiveness" is a continuous value whose range is a bounded interval. It quantifies the affiliation of each training data to the positive class. We propose a novel learning algorithm called \emph{expectation loss SVM} (e-SVM) that is devoted to the problems where only the ``positiveness" instead of a binary label of each training sample is available. Our e-SVM algorithm can also be readily extended to learn segment classifiers under weak supervision where the exact positiveness value of each training example is unobserved. In experiments, we show that the e-SVM algorithm can effectively address the segment proposal classification task under both strong supervision (e.g. the pixel-level annotations are available) and the weak supervision (e.g. only bounding-box annotations are available), and outperforms the alternative approaches. Besides, we further validate this method on two major tasks of computer vision: semantic segmentation and object detection. Our method achieves the state-of-the-art object detection performance on PASCAL VOC 2007 dataset."

Basic decisions, such as judging a person as a friend or foe, involve categorizing novel stimuli. Recent work finds that people’s category judgments are guided by a small set of examples that are retrieved from memory at decision time. This limited and stochastic retrieval places limits on human performance for probabilistic classification decisions. In light of this capacity limitation, recent work finds that idealizing training items, such that the saliency of ambiguous cases is reduced, improves human performance on novel test items. One shortcoming of previous work in idealization is that category distributions were idealized in an ad hoc or heuristic fashion. In this contribution, we take a first principles approach to constructing idealized training sets. We apply a machine teaching procedure to a cognitive model that is either limited capacity (as humans are) or unlimited capacity (as most machine learning systems are). As predicted, we find that the machine teacher recommends idealized training sets. We also find that human learners perform best when training recommendations from the machine teacher are based on a limited-capacity model. As predicted, to the extent that the learning model used by the machine teacher conforms to the true nature of human learners, the recommendations of the machine teacher prove effective. Our results provide a normative basis (given capacity constraints) for idealization procedures and offer a novel selection procedure for models of human learning.