We obtain a tight distribution-specific characterization of the sample complexity of large-margin classification with L2 regularization: We introduce the margin-adapted dimension, which is a simple function of the second order statistics of the data distribution, and show distribution-specific upper and lower bounds on the sample complexity, both governed by the margin-adapted dimension of the data distribution. The upper bounds are universal, and the lower bounds hold for the rich family of sub-Gaussian distributions with independent features. We conclude that this new quantity tightly characterizes the true sample complexity of large-margin classification. To prove the lower bound, we develop several new tools of independent interest. These include new connections between shattering and hardness of learning, new properties of shattering with linear classifiers, and a new lower bound on the smallest eigenvalue of a random Gram matrix generated by sub-Gaussian variables. Our results can be used to quantitatively compare large margin learning to other learning rules, and to improve the effectiveness of methods that use sample complexity bounds, such as active learning.

We introduce GP-FNARX: a new model for nonlinear system identification based on a nonlinear autoregressive exogenous model (NARX) with filtered regressors (F) where the nonlinear regression problem is tackled using sparse Gaussian processes (GP). We integrate data pre-processing with system identification into a fully automated procedure that goes from raw data to an identified model. Both pre-processing parameters and GP hyper-parameters are tuned by maximizing the marginal likelihood of the probabilistic model. We obtain a Bayesian model of the system's dynamics which is able to report its uncertainty in regions where the data is scarce. The automated approach, the modeling of uncertainty and its relatively low computational cost make of GP-FNARX a good candidate for applications in robotics and adaptive control.

The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. This justifies application of constraint-based causal search algorithms such as the BuildPureClusters algorithm (Silva et al. 2006) for discovering the causal structure of latent variable models to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables.

We consider the problem of naming objects in complex, natural scenes containing widely varying object appearance and subtly different names. Informed by cognitive research, we propose an approach based on sharing context based object hypotheses between visual and lexical spaces. To this end, we present the Visual Semantic Integration Model (VSIM) that represents object labels as entities shared between semantic and visual contexts and infers a new image by updating labels through context switching. At the core of VSIM is a semantic Pachinko Allocation Model and a visual nearest neighbor Latent Dirichlet Allocation Model. For inference, we derive an iterative Data Augmentation algorithm that pools the label probabilities and maximizes the joint label posterior of an image.

We provide a formulation for Local Support Vector Machines (LSVMs) that generalizes previous formulations, and brings out the explicit connections to local polynomial learning used in nonparametric estimation literature. We investigate the simplest type of LSVMs called Local Linear Support Vector Machines (LLSVMs). For the first time we establish conditions under which LLSVMs make Bayes consistent predictions at each test point $x_0$. We also establish rates at which the local risk of LLSVMs converges to the minimum value of expected local risk at each point $x_0$. Using stability arguments we establish generalization error bounds for LLSVMs.

Optimizing the acquisition matrix is useful for compressed sensing of signals that are sparse in overcomplete dictionaries, because the acquisition matrix can be adapted to the particular correlations of the dictionary atoms. In this paper a novel formulation of the optimization problem is proposed, in the form of a rank-constrained nearest correlation matrix problem. Furthermore, improvements for three existing optimization algorithms are introduced, which are shown to be particular instances of the proposed formulation. Simulation results show notable improvements and superior robustness in sparse signal recovery. Keywords: acquisition, compressed sensing, nearest correlation matrix, optimization 1. Introduction Compressed Sensing (CS) [1] studies the possibility of acquiring a signal x that is a priori known to be sparse in some dictionary D with fewer linear measurements than required by the traditional sampling theorem. In many cases the dictionary D is an orthogonal basis, but we consider here the general case of an overcomplete dictionary.

In this article we discuss some of the consequences of the mixed membership perspective on time series analysis. In its most abstract form, a mixed membership model aims to associate an individual entity with some set of attributes based on a collection of observed data. Although much of the literature on mixed membership models considers the setting in which exchangeable collections of data are associated with each member of a set of entities, it is equally natural to consider problems in which an entire time series is viewed as an entity and the goal is to characterize the time series in terms of a set of underlying dynamic attributes or "dynamic regimes". Indeed, this perspective is already present in the classical hidden Markov model, where the dynamic regimes are referred to as "states", and the collection of states realized in a sample path of the underlying process can be viewed as a mixed membership characterization of the observed time series. Our goal here is to review some of the richer modeling possibilities for time series that are provided by recent developments in the mixed membership framework.

Qualitative spatial and temporal reasoning is based on so-called qualitative calculi. Algebraic properties of these calculi have several implications on reasoning algorithms. But what exactly is a qualitative calculus? And to which extent do the qualitative calculi proposed meet these demands? The literature provides various answers to the first question but only few facts about the second. In this paper we identify the minimal requirements to binary spatio-temporal calculi and we discuss the relevance of the according axioms for representation and reasoning. We also analyze existing qualitative calculi and provide a classification involving different notions of a relation algebra.

Decomposing tensors into orthogonal factors is a well-known task in statistics, machine learning, and signal processing. We study orthogonal outer product decompositions where the factors in the summands in the decomposition are required to be orthogonal across summands, by relating this orthogonal decomposition to the singular value decompositions of the flattenings. We show that it is a non-trivial assumption for a tensor to have such an orthogonal decomposition, and we show that it is unique (up to natural symmetries) in case it exists, in which case we also demonstrate how it can be efficiently and reliably obtained by a sequence of singular value decompositions. We demonstrate how the factoring algorithm can be applied for parameter identification in latent variable and mixture models.

Restricted Boltzmann Machines (RBMs) are generative models which can learn useful representations from samples of a dataset in an unsupervised fashion. They have been widely employed as an unsupervised pre-training method in machine learning. RBMs have been modified to model time series in two main ways: The Temporal RBM stacks a number of RBMs laterally and introduces temporal dependencies between the hidden layer units; The Conditional RBM, on the other hand, considers past samples of the dataset as a conditional bias and learns a representation which takes these into account. Here we propose a new training method for both the TRBM and the CRBM, which enforces the dynamic structure of temporal datasets. We do so by treating the temporal models as denoising autoencoders, considering past frames of the dataset as corrupted versions of the present frame and minimizing the reconstruction error of the present data by the model. We call this approach Temporal Autoencoding. This leads to a significant improvement in the performance of both models in a filling-in-frames task across a number of datasets. The error reduction for motion capture data is 56\% for the CRBM and 80\% for the TRBM. Taking the posterior mean prediction instead of single samples further improves the model's estimates, decreasing the error by as much as 91\% for the CRBM on motion capture data. We also trained the model to perform forecasting on a large number of datasets and have found TA pretraining to consistently improve the performance of the forecasts. Furthermore, by looking at the prediction error across time, we can see that this improvement reflects a better representation of the dynamics of the data as opposed to a bias towards reconstructing the observed data on a short time scale.