Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and representation matrix are assumed to be known exactly in advance. However, uncertainties exist due to sampling distortion, finite grids of the parameter space of dictionary, etc. In this paper, we take a generalized sparse signal model, which simultaneously considers the sampling and representation matrix uncertainties. Based on the new signal model, a new optimization model for robust sparse signal reconstruction is proposed. This optimization model can be deduced with stochastic robust approximation analysis. Both convex relaxation and greedy algorithms are used to solve the optimization problem. For the convex relaxation method, a sufficient condition for recovery by convex relaxation is given; For the greedy algorithm, it is realized by the introduction of a pre-processing of the sensing matrix and the measurements. In numerical experiments, both simulated data and real-life ECG data based results show that the proposed method has a better performance than the current methods.

The true online TD({\lambda}) algorithm has recently been proposed (van Seijen and Sutton, 2014) as a universal replacement for the popular TD({\lambda}) algorithm, in temporal-difference learning and reinforcement learning. True online TD({\lambda}) has better theoretical properties than conventional TD({\lambda}), and the expectation is that it also results in faster learning. In this paper, we put this hypothesis to the test. Specifically, we compare the performance of true online TD({\lambda}) with that of TD({\lambda}) on challenging examples, random Markov reward processes, and a real-world myoelectric prosthetic arm. We use linear function approximation with tabular, binary, and non-binary features. We assess the algorithms along three dimensions: computational cost, learning speed, and ease of use. Our results confirm the strength of true online TD({\lambda}): 1) for sparse feature vectors, the computational overhead with respect to TD({\lambda}) is minimal; for non-sparse features the computation time is at most twice that of TD({\lambda}), 2) across all domains/representations the learning speed of true online TD({\lambda}) is often better, but never worse than that of TD({\lambda}), and 3) true online TD({\lambda}) is easier to use, because it does not require choosing between trace types, and it is generally more stable with respect to the step-size. Overall, our results suggest that true online TD({\lambda}) should be the first choice when looking for an efficient, general-purpose TD method.

In this paper, we build an organization of high-dimensional datasets that cannot be cleanly embedded into a low-dimensional representation due to missing entries and a subset of the features being irrelevant to modeling functions of interest. Our algorithm begins by defining coarse neighborhoods of the points and defining an expected empirical function value on these neighborhoods. We then generate new non-linear features with deep net representations tuned to model the approximate function, and re-organize the geometry of the points with respect to the new representation. Finally, the points are locally z-scored to create an intrinsic geometric organization which is independent of the parameters of the deep net, a geometry designed to assure smoothness with respect to the empirical function. We examine this approach on data from the Center for Medicare and Medicaid Services Hospital Quality Initiative, and generate an intrinsic low-dimensional organization of the hospitals that is smooth with respect to an expert driven function of quality.

We introduce Natural Neural Networks, a novel family of algorithms that speed up convergence by adapting their internal representation during training to improve conditioning of the Fisher matrix. In particular, we show a specific example that employs a simple and efficient reparametrization of the neural network weights by implicitly whitening the representation obtained at each layer, while preserving the feed-forward computation of the network. Such networks can be trained efficiently via the proposed Projected Natural Gradient Descent algorithm (PRONG), which amortizes the cost of these reparametrizations over many parameter updates and is closely related to the Mirror Descent online learning algorithm. We highlight the benefits of our method on both unsupervised and supervised learning tasks, and showcase its scalability by training on the large-scale ImageNet Challenge dataset.

Cross-validation (CV) is one of the main tools for performance estimation and parameter tuning in machine learning. The general recipe for computing CV estimate is to run a learning algorithm separately for each CV fold, a computationally expensive process. In this paper, we propose a new approach to reduce the computational burden of CV-based performance estimation. As opposed to all previous attempts, which are specific to a particular learning model or problem domain, we propose a general method applicable to a large class of incremental learning algorithms, which are uniquely fitted to big data problems. In particular, our method applies to a wide range of supervised and unsupervised learning tasks with different performance criteria, as long as the base learning algorithm is incremental. We show that the running time of the algorithm scales logarithmically, rather than linearly, in the number of CV folds. Furthermore, the algorithm has favorable properties for parallel and distributed implementation. Experiments with state-of-the-art incremental learning algorithms confirm the practicality of the proposed method.

This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by the lasso. We present the Condition-on-Selection method that allows for valid selective inference, and study its application to the lasso, and several other selection algorithms. In the second part, we consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parametrization of the model. We provide conditions under which our estimator is model selection consistent in the high-dimensional regime.

We present a general regularization-based framework for Multi-task learning (MTL), in which the similarity between tasks can be learned or refined using $\ell_p$-norm Multiple Kernel learning (MKL). Based on this very general formulation (including a general loss function), we derive the corresponding dual formulation using Fenchel duality applied to Hermitian matrices. We show that numerous established MTL methods can be derived as special cases from both, the primal and dual of our formulation. Furthermore, we derive a modern dual-coordinate descend optimization strategy for the hinge-loss variant of our formulation and provide convergence bounds for our algorithm. As a special case, we implement in C++ a fast LibLinear-style solver for $\ell_p$-norm MKL. In the experimental section, we analyze various aspects of our algorithm such as predictive performance and ability to reconstruct task relationships on biologically inspired synthetic data, where we have full control over the underlying ground truth. We also experiment on a new dataset from the domain of computational biology that we collected for the purpose of this paper. It concerns the prediction of transcription start sites (TSS) over nine organisms, which is a crucial task in gene finding. Our solvers including all discussed special cases are made available as open-source software as part of the SHOGUN machine learning toolbox (available at \url{http://shogun.ml}).

Factorized Information Criterion (FIC) is a recently developed information criterion, based on which a novel model selection methodology, namely Factorized Asymptotic Bayesian (FAB) Inference, has been developed and successfully applied to various hierarchical Bayesian models. The Dirichlet Process (DP) prior, and one of its well known representations, the Chinese Restaurant Process (CRP), derive another line of model selection methods. FIC can be viewed as a prior distribution over the latent variable configurations. Under this view, we prove that when the parameter dimensionality $D_{c}=2$, FIC is equivalent to CRP. We argue that when $D_{c}>2$, FIC avoids an inherent problem of DP/CRP, i.e. the data likelihood will dominate the impact of the prior, and thus the model selection capability will weaken as $D_{c}$ increases. However, FIC overestimates the data likelihood. As a result, FIC may be overly biased towards models with less components. We propose a natural generalization of FIC, which finds a middle ground between CRP and FIC, and may yield more accurate model selection results than FIC.

This paper proposes a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform confidence subgraph. Due to the presence of unknown marginal transformations, we propose a pseudo likelihood based inferential approach. In sharp contrast to the existing high dimensional score test method, our method is free of tuning parameters given an initial estimator, and extends the scope of the existing likelihood based inferential framework. Furthermore, we propose a U-statistic multiplier bootstrap method to construct the confidence subgraph. We show that the constructed subgraph is contained in the true graph with probability greater than a given nominal level. Compared with existing methods for constructing confidence subgraphs, our method does not rely on Gaussian or sub-Gaussian assumptions. The theoretical properties of the proposed inferential methods are verified by thorough numerical experiments and real data analysis.

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved sub-problems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed Divide-and-Conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-Conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging sub-problems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem.