We propose a novel method for multiple clustering that assumes a co-clustering structure (partitions in both rows and columns of the data matrix) in each view. The new method is applicable to high-dimensional data. It is based on a nonparametric Bayesian approach in which the number of views and the number of feature-/subject clusters are inferred in a data-driven manner. We simultaneously model different distribution families, such as Gaussian, Poisson, and multinomial distributions in each cluster block. This makes our method applicable to datasets consisting of both numerical and categorical variables, which biomedical data typically do. Clustering solutions are based on variational inference with mean field approximation. We apply the proposed method to synthetic and real data, and show that our method outperforms other multiple clustering methods both in recovering true cluster structures and in computation time. Finally, we apply our method to a depression dataset with no true cluster structure available, from which useful inferences are drawn about possible clustering structures of the data.

We propose a stochastic gradient Markov chain Monte Carlo (SG-MCMC) algorithm for scalable inference in mixed-membership stochastic blockmodels (MMSB). Our algorithm is based on the stochastic gradient Riemannian Langevin sampler and achieves both faster speed and higher accuracy at every iteration than the current state-of-the-art algorithm based on stochastic variational inference. In addition we develop an approximation that can handle models that entertain a very large number of communities. The experimental results show that SG-MCMC strictly dominates competing algorithms in all cases.

A good measure of similarity between data points is crucial to many tasks in machine learning. Similarity and metric learning methods learn such measures automatically from data, but they do not scale well respect to the dimensionality of the data. In this paper, we propose a method that can learn efficiently similarity measure from high-dimensional sparse data. The core idea is to parameterize the similarity measure as a convex combination of rank-one matrices with specific sparsity structures. The parameters are then optimized with an approximate Frank-Wolfe procedure to maximally satisfy relative similarity constraints on the training data. Our algorithm greedily incorporates one pair of features at a time into the similarity measure, providing an efficient way to control the number of active features and thus reduce overfitting. It enjoys very appealing convergence guarantees and its time and memory complexity depends on the sparsity of the data instead of the dimension of the feature space. Our experiments on real-world high-dimensional datasets demonstrate its potential for classification, dimensionality reduction and data exploration.

The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix (of magnitude $\lambda$ and size $k_m \times k_n$) contaminated with a noise matrix (of size $m \times n$), we establish two transition thresholds for the signal to noise $\lambda/\sigma$ ratio in terms of $m$, $n$, $k_m$, and $k_n$. The first threshold, $\sf SNR_c$, corresponds to the computational boundary. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the \textit{hidden clique hypothesis}. We introduce adaptive linear time spectral algorithms that identify the submatrix with high probability when the signal strength is above the threshold $\sf SNR_c$. The second threshold, $\sf SNR_s$, captures the statistical boundary, below which no method can succeed with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. The results show an interesting phenomenon that $\sf SNR_c$ is always significantly larger than $\sf SNR_s$, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization.

The application of existing methods for constructing optimal dynamic treatment regimes is limited to cases where investigators are interested in optimizing a utility function over a fixed period of time (finite horizon). In this manuscript, we develop an inferential procedure based on temporal difference residuals for optimal dynamic treatment regimes in infinite-horizon settings, where there is no a priori fixed end of follow-up point. The proposed method can be used to determine the optimal regime in chronic diseases where patients are monitored and treated throughout their life. We derive large sample results necessary for conducting inference. We also simulate a cohort of patients with diabetes to mimic the third wave of the National Health and Nutrition Examination Survey, and we examine the performance of the proposed method in controlling the level of hemoglobin A1c. Supplementary materials for this article are available online.

Acyclic digraphs are the underlying representation of Bayesian networks, a widely used class of probabilistic graphical models. Learning the underlying graph from data is a way of gaining insights about the structural properties of a domain. Structure learning forms one of the inference challenges of statistical graphical models. MCMC methods, notably structure MCMC, to sample graphs from the posterior distribution given the data are probably the only viable option for Bayesian model averaging. Score modularity and restrictions on the number of parents of each node allow the graphs to be grouped into larger collections, which can be scored as a whole to improve the chain's convergence. Current examples of algorithms taking advantage of grouping are the biased order MCMC, which acts on the alternative space of permuted triangular matrices, and non ergodic edge reversal moves. Here we propose a novel algorithm, which employs the underlying combinatorial structure of DAGs to define a new grouping. As a result convergence is improved compared to structure MCMC, while still retaining the property of producing an unbiased sample. Finally the method can be combined with edge reversal moves to improve the sampler further.

Principal component analysis (PCA) for binary data, known as logistic PCA, has become a popular alternative to dimensionality reduction of binary data. It is motivated as an extension of ordinary PCA by means of a matrix factorization, akin to the singular value decomposition, that maximizes the Bernoulli log-likelihood. We propose a new formulation of logistic PCA which extends Pearson's formulation of a low dimensional data representation with minimum error to binary data. Our formulation does not require a matrix factorization, as previous methods do, but instead looks for projections of the natural parameters from the saturated model. Due to this difference, the number of parameters does not grow with the number of observations and the principal component scores on new data can be computed with simple matrix multiplication. We derive explicit solutions for data matrices of special structure and provide computationally efficient algorithms for solving for the principal component loadings. Through simulation experiments and an analysis of medical diagnoses data, we compare our formulation of logistic PCA to the previous formulation as well as ordinary PCA to demonstrate its benefits.

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results.

As part of daily monitoring of human activities, wearable sensors and devices are becoming increasingly popular sources of data. With the advent of smartphones equipped with acceloremeter, gyroscope and camera; it is now possible to develop activity classification platforms everyone can use conveniently. In this paper, we propose a fast inference method for an unsupervised non-parametric time series model namely variational inference for sticky HDP-SLDS(Hierarchical Dirichlet Process Switching Linear Dynamical System). We show that the proposed algorithm can differentiate various indoor activities such as sitting, walking, turning, going up/down the stairs and taking the elevator using only the acceloremeter of an Android smartphone Samsung Galaxy S4. We used the front camera of the smartphone to annotate activity types precisely. We compared the proposed method with Hidden Markov Models with Gaussian emission probabilities on a dataset of 10 subjects. We showed that the efficacy of the stickiness property. We further compared the variational inference to the Gibbs sampler on the same model and show that variational inference is faster in one order of magnitude.

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering.