In many healthcare settings, intuitive decision rules for risk stratification can help effective hospital resource allocation. This paper introduces a novel variant of decision tree algorithms that produces a chain of decisions, not a general tree. Our algorithm, $\alpha$-Carving Decision Chain (ACDC), sequentially carves out "pure" subsets of the majority class examples. The resulting chain of decision rules yields a pure subset of the minority class examples. Our approach is particularly effective in exploring large and class-imbalanced health datasets. Moreover, ACDC provides an interactive interpretation in conjunction with visual performance metrics such as Receiver Operating Characteristics curve and Lift chart.

Databases in domains such as healthcare are routinely released to the public in aggregated form. Unfortunately, naive modeling with aggregated data may significantly diminish the accuracy of inferences at the individual level. This paper addresses the scenario where features are provided at the individual level, but the target variables are only available as histogram aggregates or order statistics. We consider a limiting case of generalized linear modeling when the target variables are only known up to permutation, and explore how this relates to permutation testing; a standard technique for assessing statistical dependency. Based on this relationship, we propose a simple algorithm to estimate the model parameters and individual level inferences via alternating imputation and standard generalized linear model fitting. Our results suggest the effectiveness of the proposed approach when, in the original data, permutation testing accurately ascertains the veracity of the linear relationship. The framework is extended to general histogram data with larger bins - with order statistics such as the median as a limiting case. Our experimental results on simulated data and aggregated healthcare data suggest a diminishing returns property with respect to the granularity of the histogram - when a linear relationship holds in the original data, the targets can be predicted accurately given relatively coarse histograms.

Learning the true ordering between objects by aggregating a set of expert opinion rank order lists is an important and ubiquitous problem in many applications ranging from social choice theory to natural language processing and search aggregation. We study the problem of unsupervised rank aggregation where no ground truth ordering information in available, neither about the true preference ordering between any set of objects nor about the quality of individual rank lists. Aggregating the often inconsistent and poor quality rank lists in such an unsupervised manner is a highly challenging problem, and standard consensus-based methods are often ill-defined, and difficult to solve. In this manuscript we propose a novel framework to bypass these issues by using object attributes to augment the standard rank aggregation framework. We design algorithms that learn joint models on both rank lists and object features to obtain an aggregated rank ordering that is more accurate and robust, and also helps weed out rank lists of dubious validity. We validate our techniques on synthetic datasets where our algorithm is able to estimate the true rank ordering even when the rank lists are corrupted. Experiments on three real datasets, MQ2008, MQ2008 and OHSUMED, show that using object features can result in significant improvement in performance over existing rank aggregation methods that do not use object information. Furthermore, when at least some of the rank lists are of high quality, our methods are able to effectively exploit their high expertise to output an aggregated rank ordering of great accuracy.

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.

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.

A gamma process dynamic Poisson factor analysis model is proposed to factorize a dynamic count matrix, whose columns are sequentially observed count vectors. The model builds a novel Markov chain that sends the latent gamma random variables at time $(t-1)$ as the shape parameters of those at time $t$, which are linked to observed or latent counts under the Poisson likelihood. The significant challenge of inferring the gamma shape parameters is fully addressed, using unique data augmentation and marginalization techniques for the negative binomial distribution. The same nonparametric Bayesian model also applies to the factorization of a dynamic binary matrix, via a Bernoulli-Poisson link that connects a binary observation to a latent count, with closed-form conditional posteriors for the latent counts and efficient computation for sparse observations. We apply the model to text and music analysis, with state-of-the-art results.

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low--rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin--tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low--rank, such as block--sparsity, or a superposition structure of low--rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of exponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized $M$--estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.

We present a general framework for constructing prior distributions with structured variables. The prior is defined as the information projection of a base distribution onto distributions supported on the constraint set of interest. In cases where this projection is intractable, we propose a family of parameterized approximations indexed by subsets of the domain. We further analyze the special case of sparse structure. While the optimal prior is intractable in general, we show that approximate inference using convex subsets is tractable, and is equivalent to maximizing a submodular function subject to cardinality constraints. As a result, inference using greedy forward selection provably achieves within a factor of (1-1/e) of the optimal objective value. Our work is motivated by the predictive modeling of high-dimensional functional neuroimaging data. For this task, we employ the Gaussian base distribution induced by local partial correlations and consider the design of priors to capture the domain knowledge of sparse support. Experimental results on simulated data and high dimensional neuroimaging data show the effectiveness of our approach in terms of support recovery and predictive accuracy.

Transposable data represents interactions among two sets of entities, and are typically represented as a matrix containing the known interaction values. Additional side information may consist of feature vectors specific to entities corresponding to the rows and/or columns of such a matrix. Further information may also be available in the form of interactions or hierarchies among entities along the same mode (axis). We propose a novel approach for modeling transposable data with missing interactions given additional side information. The interactions are modeled as noisy observations from a latent noise free matrix generated from a matrix-variate Gaussian process. The construction of row and column covariances using side information provides a flexible mechanism for specifying a-priori knowledge of the row and column correlations in the data. Further, the use of such a prior combined with the side information enables predictions for new rows and columns not observed in the training data. In this work, we combine the matrix-variate Gaussian process model with low rank constraints. The constrained Gaussian process approach is applied to the prediction of hidden associations between genes and diseases using a small set of observed associations as well as prior covariances induced by gene-gene interaction networks and disease ontologies. The proposed approach is also applied to recommender systems data which involves predicting the item ratings of users using known associations as well as prior covariances induced by social networks. We present experimental results that highlight the performance of constrained matrix-variate Gaussian process as compared to state of the art approaches in each domain.

We present a novel approach for constrained Bayesian inference. Unlike current methods, our approach does not require convexity of the constraint set. We reduce the constrained variational inference to a parametric optimization over the feasible set of densities and propose a general recipe for such problems. We apply the proposed constrained Bayesian inference approach to multitask learning subject to rank constraints on the weight matrix. Further, constrained parameter estimation is applied to recover the sparse conditional independence structure encoded by prior precision matrices. Our approach is motivated by reverse inference for high dimensional functional neuroimaging, a domain where the high dimensionality and small number of examples requires the use of constraints to ensure meaningful and effective models. For this application, we propose a model that jointly learns a weight matrix and the prior inverse covariance structure between different tasks. We present experimental validation showing that the proposed approach outperforms strong baseline models in terms of predictive performance and structure recovery.