We consider the computational complexity of probabilistic inference in Latent Dirichlet Allocation (LDA). First, we study the problem of finding the maximum a posteriori (MAP) assignment of topics to words, where the document's topic distribution is integrated out. We show that, when the effective number of topics per document is small, exact inference takes polynomial time. In contrast, we show that, when a document has a large number of topics, finding the MAP assignment of topics to words in LDA is NP-hard. Next, we consider the problem of finding the MAP topic distribution for a document, where the topic-word assignments are integrated out. We show that this problem is also NP-hard. Finally, we briefly discuss the problem of sampling from the posterior, showing that this is NP-hard in one restricted setting, but leaving open the general question.

An increasing number of experimental studies indicate that perception encodes a posterior probability distribution over possible causes of sensory stimuli, which is used to act close to optimally in the environment. One outstanding difficulty with this hypothesis is that the exact posterior will in general be too complex to be represented directly, and thus neurons will have to represent an approximation of this distribution. Two influential proposals of efficient posterior representation by neural populations are: 1) neural activity represents samples of the underlying distribution, or 2) they represent a parametric representation of a variational approximation of the posterior. We show that these approaches can be combined for an inference scheme that retains the advantages of both: it is able to represent multiple modes and arbitrary correlations, a feature of sampling methods, and it reduces the represented space to regions of high probability mass, a strength of variational approximations. Neurally, the combined method can be interpreted as a feed-forward preselection of the relevant state space, followed by a neural dynamics implementation of Markov Chain Monte Carlo (MCMC) to approximate the posterior over the relevant states. We demonstrate the effectiveness and efficiency of this approach on a sparse coding model. In numerical experiments on artificial data and image patches, we compare the performance of the algorithms to that of exact EM, variational state space selection alone, MCMC alone, and the combined select and sample approach. The select and sample approach integrates the advantages of the sampling and variational approximations, and forms a robust, neurally plausible, and very efficient model of processing and learning in cortical networks. For sparse coding we show applications easily exceeding a thousand observed and a thousand hidden dimensions.

Latent variable models are frequently used to identify structure in dichotomous network data, in part because they give rise to a Bernoulli product likelihood that is both well understood and consistent with the notion of exchangeable random graphs. In this article we propose conservative confidence sets that hold with respect to these underlying Bernoulli parameters as a function of any given partition of network nodes, enabling us to assess estimates of \emph{residual} network structure, that is, structure that cannot be explained by known covariates and thus cannot be easily verified by manual inspection. We demonstrate the proposed methodology by analyzing student friendship networks from the National Longitudinal Survey of Adolescent Health that include race, gender, and school year as covariates. We employ a stochastic expectation-maximization algorithm to fit a logistic regression model that includes these explanatory variables as well as a latent stochastic blockmodel component and additional node-specific effects. Although maximum-likelihood estimates do not appear consistent in this context, we are able to evaluate confidence sets as a function of different blockmodel partitions, which enables us to qualitatively assess the significance of estimated residual network structure relative to a baseline, which models covariates but lacks block structure.

Simultaneous recordings of many neurons embedded within a recurrently-connected cortical network may provide concurrent views into the dynamical processes of that network, and thus its computational function. In principle, these dynamics might be identified by purely unsupervised, statistical means. Here, we show that a Hidden Switching Linear Dynamical Systems (HSLDS) model---in which multiple linear dynamical laws approximate a nonlinear and potentially non-stationary dynamical process---is able to distinguish different dynamical regimes within single-trial motor cortical activity associated with the preparation and initiation of hand movements. The regimes are identified without reference to behavioural or experimental epochs, but nonetheless transitions between them correlate strongly with external events whose timing may vary from trial to trial. The HSLDS model also performs better than recent comparable models in predicting the firing rate of an isolated neuron based on the firing rates of others, suggesting that it captures more of the "shared variance" of the data. Thus, the method is able to trace the dynamical processes underlying the coordinated evolution of network activity in a way that appears to reflect its computational role.

The difficulty in inverse reinforcement learning (IRL) arises in choosing the best reward function since there are typically an infinite number of reward functions that yield the given behaviour data as optimal. Using a Bayesian framework, we address this challenge by using the maximum a posteriori (MAP) estimation for the reward function, and show that most of the previous IRL algorithms can be modeled into our framework. We also present a gradient method for the MAP estimation based on the (sub)differentiability of the posterior distribution. We show the effectiveness of our approach by comparing the performance of the proposed method to those of the previous algorithms.

We propose a novel generative model that is able to reason jointly about the 3D scene layout as well as the 3D location and orientation of objects in the scene. In particular, we infer the scene topology, geometry as well as traffic activities from a short video sequence acquired with a single camera mounted on a moving car. Our generative model takes advantage of dynamic information in the form of vehicle tracklets as well as static information coming from semantic labels and geometry (i.e., vanishing points). Experiments show that our approach outperforms a discriminative baseline based on multiple kernel learning (MKL) which has access to the same image information. Furthermore, as we reason about objects in 3D, we are able to significantly increase the performance of state-of-the-art object detectors in their ability to estimate object orientation.

Compositional models provide an elegant formalism for representing the visual appearance of highly variable objects. While such models are appealing from a theoretical point of view, it has been difficult to demonstrate that they lead to performance advantages on challenging datasets. Here we develop a grammar model for person detection and show that it outperforms previous high-performance systems on the PASCAL benchmark. Our model represents people using a hierarchy of deformable parts, variable structure and an explicit model of occlusion for partially visible objects. To train the model, we introduce a new discriminative framework for learning structured prediction models from weakly-labeled data.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy. Furthermore, VT converges faster and results in sparser (simpler) models, thus realizing an automatic Occam's razor for HMM learning. For more general scenario VT can be worse compared to ML but still capable of correctly recovering most of the parameters.

We introduce the Piecewise-Constant Conditional Intensity Model, a model for learning temporal dependencies in event streams. We describe a closed-form Bayesian approach to learning these models, and describe an importance sampling algorithm for forecasting future events using these models, using a proposal distribution based on Poisson superposition. We then use synthetic data, supercomputer event logs, and web search query logs to illustrate that our learning algorithm can efficiently learn nonlinear temporal dependencies, and that our importance sampling algorithm can effectively forecast future events.

For most scene understanding tasks (such as object detection or depth estimation), the classifiers need to consider contextual information in addition to the local features. We can capture such contextual information by taking as input the features/attributes from all the regions in the image. However, this contextual dependence also varies with the spatial location of the region of interest, and we therefore need a different set of parameters for each spatial location. This results in a very large number of parameters. In this work, we model the independence properties between the parameters for each location and for each task, by defining a Markov Random Field (MRF) over the parameters. In particular, two sets of parameters are encouraged to have similar values if they are spatially close or semantically close. Our method is, in principle, complementary to other ways of capturing context such as the ones that use a graphical model over the labels instead. In extensive evaluation over two different settings, of multi-class object detection and of multiple scene understanding tasks (scene categorization, depth estimation, geometric labeling), our method beats the state-of-the-art methods in all the four tasks.