The stochastic linear bandit problem proceeds in rounds where at each round the algorithm selects a vector from a decision set after which it receives a noisy linear loss parameterized by an unknown vector. The goal in such a problem is to minimize the (pseudo) regret which is the difference between the total expected loss of the algorithm and the total expected loss of the best fixed vector in hindsight. In this paper, we consider settings where the unknown parameter has structure, e.g., sparse, group sparse, low-rank, which can be captured by a norm, e.g., $L_1$, $L_{(1,2)}$, nuclear norm. We focus on constructing confidence ellipsoids which contain the unknown parameter across all rounds with high-probability. We show the radius of such ellipsoids depend on the Gaussian width of sets associated with the norm capturing the structure. Such characterization leads to tighter confidence ellipsoids and, therefore, sharper regret bounds compared to bounds in the existing literature which are based on the ambient dimensionality.

Statistical modeling of local precipitation involves understanding local, regional and global factors informative of precipitation variability in a region. Modern machine learning methods for feature selection can potentially be explored for identifying statistically significant features from pool of potential predictors of precipitation. In this work, we consider sparse regression, which simultaneously performs feature selection and regression, followed by random permutation tests for selecting dominant factors. We consider average winter precipitation over Great Lakes Region in order to identify its dominant influencing factors.Experiments show that global climate indices, computed at different temporal lags, offer predictive information for winter precipitation. Further, among the dominant factors identified using randomized permutation tests, multiple climate indices indicate the influence of geopotential height patterns on winter precipitation.Using composite analysis, we illustrate that certain patterns are indeed typical in high and low precipitation years, and offer plausible scientific reasons for variations in precipitation.Thus, feature selection methods can be useful in identifying influential climate processes and variables, and thereby provide useful hypotheses over physical mechanisms affecting local precipitation.

In recent years, structured matrix recovery problems have gained considerable attention for its real world applications, such as recommender systems and computer vision. Much of the existing work has focused on matrices with low-rank structure, and limited progress has been made matrices with other types of structure. In this paper we present non-asymptotic analysis for estimation of generally structured matrices via the generalized Dantzig selector under generic sub-Gaussian measurements. We show that the estimation error can always be succinctly expressed in terms of a few geometric measures of suitable sets which only depend on the structure of the underlying true matrix. In addition, we derive the general bounds on these geometric measures for structures characterized by unitarily invariant norms, which is a large family covering most matrix norms of practical interest. Examples are provided to illustrate the utility of our theoretical development.

In this paper, we present a unified analysis of matrix completion under general low-dimensional structural constraints induced by {\em any} norm regularization. We consider two estimators for the general problem of structured matrix completion, and provide unified upper bounds on the sample complexity and the estimation error. Our analysis relies on results from generic chaining, and we establish two intermediate results of independent interest: (a) in characterizing the size or complexity of low dimensional subsets in high dimensional ambient space, a certain partial complexity measure encountered in the analysis of matrix completion problems is characterized in terms of a well understood complexity measure of Gaussian widths, and (b) it is shown that a form of restricted strong convexity holds for matrix completion problems under general norm regularization. Further, we provide several non-trivial examples of structures included in our framework, notably the recently proposed spectral $k$-support norm.

Hidden semi-Markov models (HSMMs) are latent variable models which allow latent state persistence and can be viewed as a generalization of the popular hidden Markov models (HMMs). In this paper, we introduce a novel spectral algorithm to perform inference in HSMMs. Unlike expectation maximization (EM), our approach correctly estimates the probability of given observation sequence based on a set of training sequences. Our approach is based on estimating moments from the sample, whose number of dimensions depends only logarithmically on the maximum length of the hidden state persistence. Moreover, the algorithm requires only a few matrix inversions and is therefore computationally efficient. Empirical evaluations on synthetic and real data demonstrate the advantage of the algorithm over EM in terms of speed and accuracy, especially for large datasets.

While considerable advances have been made in estimating high-dimensional structured models from independent data using Lasso-type models, limited progress has been made for settings when the samples are dependent. We consider estimating structured VAR (vector auto-regressive models), where the structure can be captured by any suitable norm, e.g., Lasso, group Lasso, order weighted Lasso, sparse group Lasso, etc. In VAR setting with correlated noise, although there is strong dependence over time and covariates, we establish bounds on the non-asymptotic estimation error of structured VAR parameters. Surprisingly, the estimation error is of the same order as that of the corresponding Lasso-type estimator with independent samples, and the analysis holds for any norm. Our analysis relies on results in generic chaining, sub-exponential martingales, and spectral representation of VAR models. Experimental results on synthetic data with a variety of structures as well as real aviation data are presented, validating theoretical results.

In this work we consider the problem of anomaly detection in heterogeneous, multivariate, variable-length time series datasets. Our focus is on the aviation safety domain, where data objects are flights and time series are sensor readings and pilot switches. In this context the goal is to detect anomalous flight segments, due to mechanical, environmental, or human factors in order to identifying operationally significant events and provide insights into the flight operations and highlight otherwise unavailable potential safety risks and precursors to accidents. For this purpose, we propose a framework which represents each flight using a semi-Markov switching vector autoregressive (SMS-VAR) model. Detection of anomalies is then based on measuring dissimilarities between the model's prediction and data observation. The framework is scalable, due to the inherent parallel nature of most computations, and can be used to perform online anomaly detection. Extensive experimental results on simulated and real datasets illustrate that the framework can detect various types of anomalies along with the key parameters involved.

For structured estimation problems with atomic norms, recent advances in the literature express sample complexity and estimation error bounds in terms of certain geometric measures, in particular Gaussian width of the unit norm ball, Gaussian width of a spherical cap induced by a tangent cone, and a restricted norm compatibility constant. However, given an atomic norm, bounding these geometric measures can be difficult. In this paper, we present general upper bounds for such geometric measures, which only require simple information of the atomic norm under consideration, and we establish tightness of these bounds by providing the corresponding lower bounds. We show applications of our analysis to certain atomic norms, especially k-support norm, for which existing result is incomplete.

Matrix completion problems have been widely studied under special low dimensional structures such as low rank or structure induced by decomposable norms. In this paper, we present a unified analysis of matrix completion under general low-dimensional structural constraints induced by {\em any} norm regularization.We consider two estimators for the general problem of structured matrix completion, and provide unified upper bounds on the sample complexity and the estimation error. Our analysis relies on generic chaining, and we establish two intermediate results of independent interest: (a) in characterizing the size or complexity of low dimensional subsets in high dimensional ambient space, a certain \textit{\modified}~complexity measure encountered in the analysis of matrix completion problems is characterized in terms of a well understood complexity measure of Gaussian widths, and (b) it is shown that a form of restricted strong convexity holds for matrix completion problems under general norm regularization. Further, we provide several non-trivial examples of structures included in our framework, notably including the recently proposed spectral $k$-support norm.

We consider the problem of high-dimensional structured estimation with norm-regularized estimators, such as Lasso, when the design matrix and noise are drawn from sub-exponential distributions.Existing results only consider sub-Gaussian designs and noise, and both the sample complexity and non-asymptotic estimation error have been shown to depend on the Gaussian width of suitable sets. In contrast, for the sub-exponential setting, we show that the sample complexity and the estimation error will depend on the exponential width of the corresponding sets, and the analysis holds for any norm. Further, using generic chaining, we show that the exponential width for any set will be at most $\sqrt{\log p}$ times the Gaussian width of the set, yielding Gaussian width based results even for the sub-exponential case. Further, for certain popular estimators, viz Lasso and Group Lasso, using a VC-dimension based analysis, we show that the sample complexity will in fact be the same order as Gaussian designs. Our general analysis and results are the first in the sub-exponential setting, and are readily applicable to special sub-exponential families such as log-concave and extreme-value distributions.