State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning in nonlinear nonparametric state-space models. We place a Gaussian process prior over the transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. However, to enable efficient inference, we marginalize over the dynamics of the model and instead infer directly the joint smoothing distribution through the use of specially tailored Particle Markov Chain Monte Carlo samplers. Once an approximation of the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. We make use of sparse Gaussian process models to greatly reduce the computational complexity of the approach.

We show that either explicitly or implicitly, various well-known graph-based models exhibit a common significant \emph{harmonic} structure in its target function -- the value of a vertex is approximately the weighted average of the values of its adjacent neighbors. Understanding of such structure and analysis of the loss defined over such structure help reveal important properties of the target function over a graph. In this paper, we show that the variation of the target function across a cut can be upper and lower bounded by the ratio of its harmonic loss and the cut cost. We use this to develop an analytical tool and analyze 5 popular models in graph-based learning: absorbing random walks, partially absorbing random walks, hitting times, pseudo-inverse of graph Laplacian, and eigenvectors of the Laplacian matrices. Our analysis well explains several open questions of these models reported in the literature. Furthermore, it provides theoretical justifications and guidelines for their practical use. Simulations on synthetic and real datasets support our analysis.

Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. and Amini et al. proposed variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous theoretical results to the more canonical spectral clustering algorithm in a way that removes any assumption on the minimum degree and provides guidance on the choice of tuning parameter. Moreover, our results show how the star shape" in the eigenvectors--which are consistently observed in empirical networks--can be explained by the Degree-Corrected Stochastic Blockmodel and the Extended Planted Partition model, two statistical model that allow for highly heterogeneous degrees. Throughout, the paper characterizes and justifies several of the variations of the spectral clustering algorithm in terms of these models. "

Humans recognize visually-presented objects rapidly and accurately. To understand this ability, we seek to construct models of the ventral stream, the series of cortical areas thought to subserve object recognition. One tool to assess the quality of a model of the ventral stream is the Representation Dissimilarity Matrix (RDM), which uses a set of visual stimuli and measures the distances produced in either the brain (i.e. fMRI voxel responses, neural firing rates) or in models (features). Previous work has shown that all known models of the ventral stream fail to capture the RDM pattern observed in either IT cortex, the highest ventral area, or in the human ventral stream. In this work, we construct models of the ventral stream using a novel optimization procedure for category-level object recognition problems, and produce RDMs resembling both macaque IT and human ventral stream. The model, while novel in the optimization procedure, further develops a long-standing functional hypothesis that the ventral visual stream is a hierarchically arranged series of processing stages optimized for visual object recognition.

The recently introduced continuous Skip-gram model is an efficient method for learning high-quality distributed vector representations that capture a large number of precise syntactic and semantic word relationships. In this paper we present several improvements that make the Skip-gram model more expressive and enable it to learn higher quality vectors more rapidly. We show that by subsampling frequent words we obtain significant speedup, and also learn higher quality representations as measured by our tasks. We also introduce Negative Sampling, a simplified variant of Noise Contrastive Estimation (NCE) that learns more accurate vectors for frequent words compared to the hierarchical softmax. An inherent limitation of word representations is their indifference to word order and their inability to represent idiomatic phrases. For example, the meanings of Canada'' and "Air'' cannot be easily combined to obtain "Air Canada''. Motivated by this example, we present a simple and efficient method for finding phrases, and show that their vector representations can be accurately learned by the Skip-gram model. "

Recently, it was shown that by dropping out hidden activities with a probability of 0.5, deep neural networks can perform very well. We describe a model in which a binary belief network is overlaid on a neural network and is used to decrease the information content of its hidden units by selectively setting activities to zero. This ''dropout network can be trained jointly with the neural network by approximately computing local expectations of binary dropout variables, computing derivatives using back-propagation, and using stochastic gradient descent. Interestingly, experiments show that the learnt dropout network parameters recapitulate the neural network parameters, suggesting that a good dropout network regularizes activities according to magnitude. When evaluated on the MNIST and NORB datasets, we found our method can be used to achieve lower classification error rates than other feather learning methods, including standard dropout, denoising auto-encoders, and restricted Boltzmann machines. For example, our model achieves 5.8% error on the NORB test set, which is better than state-of-the-art results obtained using convolutional architectures. "

We present the first result for kernel regression where the procedure adapts locally at a point $x$ to both the unknown local dimension of the metric and the unknown H\{o}lder-continuity of the regression function at $x$. The result holds with high probability simultaneously at all points $x$ in a metric space of unknown structure."

One approach to computer object recognition and modeling the brain's ventral stream involves unsupervised learning of representations that are invariant to common transformations. However, applications of these ideas have usually been limited to 2D affine transformations, e.g., translation and scaling, since they are easiest to solve via convolution. In accord with a recent theory of transformation-invariance, we propose a model that, while capturing other common convolutional networks as special cases, can also be used with arbitrary identity-preserving transformations. The model's wiring can be learned from videos of transforming objects---or any other grouping of images into sets by their depicted object. Through a series of successively more complex empirical tests, we study the invariance/discriminability properties of this model with respect to different transformations. First, we empirically confirm theoretical predictions for the case of 2D affine transformations. Next, we apply the model to non-affine transformations: as expected, it performs well on face verification tasks requiring invariance to the relatively smooth transformations of 3D rotation-in-depth and changes in illumination direction. Surprisingly, it can also tolerate clutter transformations'' which map an image of a face on one background to an image of the same face on a different background. Motivated by these empirical findings, we tested the same model on face verification benchmark tasks from the computer vision literature: Labeled Faces in the Wild, PubFig and a new dataset we gathered---achieving strong performance in these highly unconstrained cases as well."

This paper addresses the problem of automatic generation of features for value function approximation in reinforcement learning. Bellman Error Basis Functions (BEBFs) have been shown to improve the error of policy evaluation with function approximation, with a convergence rate similar to that of value iteration. We propose a simple, fast and robust algorithm based on random projections, which generates BEBFs for sparse feature spaces. We provide a finite sample analysis of the proposed method, and prove that projections logarithmic in the dimension of the original space guarantee a contraction in the error. Empirical results demonstrate the strength of this method in domains in which choosing a good state representation is challenging.

In the high-dimensional regression model a response variable is linearly related to $p$ covariates, but the sample size $n$ is smaller than $p$. We assume that only a small subset of covariates is `active' (i.e., the corresponding coefficients are non-zero), and consider the model-selection problem of identifying the active covariates. A popular approach is to estimate the regression coefficients through the Lasso ($\ell_1$-regularized least squares). This is known to correctly identify the active set only if the irrelevant covariates are roughly orthogonal to the relevant ones, as quantified through the so called `irrepresentability' condition. In this paper we study the `Gauss-Lasso' selector, a simple two-stage method that first solves the Lasso, and then performs ordinary least squares restricted to the Lasso active set. We formulate `generalized irrepresentability condition' (GIC), an assumption that is substantially weaker than irrepresentability. We prove that, under GIC, the Gauss-Lasso correctly recovers the active set.