We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch method, the proposed method offers a stochastic update rule where each iteration requires only one or few sample observations. Moreover, our method can naturally afford mini-batch update and it gives speed up of convergence. We show that, under mild assumptions, our method converges exponentially. The numerical experiments show that our method actually performs efficiently.

Adaptive networks are suitable for decentralized inference tasks, e.g., to monitor complex natural phenomena. Recent research works have intensively studied distributed optimization problems in the case where the nodes have to estimate a single optimum parameter vector collaboratively. However, there are many important applications that are multitask-oriented in the sense that there are multiple optimum parameter vectors to be inferred simultaneously, in a collaborative manner, over the area covered by the network. In this paper, we employ diffusion strategies to develop distributed algorithms that address multitask problems by minimizing an appropriate mean-square error criterion with $\ell_2$-regularization. The stability and convergence of the algorithm in the mean and in the mean-square sense is analyzed. Simulations are conducted to verify the theoretical findings, and to illustrate how the distributed strategy can be used in several useful applications related to spectral sensing, target localization, and hyperspectral data unmixing.

We propose a family of multivariate Gaussian process models for correlated outputs, based on assuming that the likelihood function takes the generic form of the multivariate exponential family distribution (EFD). We denote this model as a multivariate generalized Gaussian process model, and derive Taylor and Laplace algorithms for approximate inference on the generic model. By instantiating the EFD with specific parameter functions, we obtain two novel GP models (and corresponding inference algorithms) for correlated outputs: 1) a Von-Mises GP for angle regression; and 2) a Dirichlet GP for regressing on the multinomial simplex.

High-throughput sequencing allows the detection and quantification of frequencies of somatic single nucleotide variants (SNV) in heterogeneous tumor cell populations. In some cases, the evolutionary history and population frequency of the subclonal lineages of tumor cells present in the sample can be reconstructed from these SNV frequency measurements. However, automated methods to do this reconstruction are not available and the conditions under which reconstruction is possible have not been described. We describe the conditions under which the evolutionary history can be uniquely reconstructed from SNV frequencies from single or multiple samples from the tumor population and we introduce a new statistical model, PhyloSub, that infers the phylogeny and genotype of the major subclonal lineages represented in the population of cancer cells. It uses a Bayesian nonparametric prior over trees that groups SNVs into major subclonal lineages and automatically estimates the number of lineages and their ancestry. We sample from the joint posterior distribution over trees to identify evolutionary histories and cell population frequencies that have the highest probability of generating the observed SNV frequency data. When multiple phylogenies are consistent with a given set of SNV frequencies, PhyloSub represents the uncertainty in the tumor phylogeny using a partial order plot. Experiments on a simulated dataset and two real datasets comprising tumor samples from acute myeloid leukemia and chronic lymphocytic leukemia patients demonstrate that PhyloSub can infer both linear (or chain) and branching lineages and its inferences are in good agreement with ground truth, where it is available.

A family of parsimonious shifted asymmetric Laplace mixture models is introduced. We extend the mixture of factor analyzers model to the shifted asymmetric Laplace distribution. Imposing constraints on the constitute parts of the resulting decomposed component scale matrices leads to a family of parsimonious models. An explicit two-stage parameter estimation procedure is described, and the Bayesian information criterion and the integrated completed likelihood are compared for model selection. This novel family of models is applied to real data, where it is compared to its Gaussian analogue within clustering and classification paradigms.

This paper studies iteration convergence of Kronecker graphical lasso (KGLasso) algorithms for estimating the covariance of an i.i.d. Gaussian random sample under a sparse Kronecker-product covariance model and MSE convergence rates. The KGlasso model, originally called the transposable regularized covariance model by Allen ["Transposable regularized covariance models with an application to missing data imputation," Ann. Appl. Statist., vol. 4, no. 2, pp. 764-790, 2010], implements a pair of $\ell_1$ penalties on each Kronecker factor to enforce sparsity in the covariance estimator. The KGlasso algorithm generalizes Glasso, introduced by Yuan and Lin ["Model selection and estimation in the Gaussian graphical model," Biometrika, vol. 94, pp. 19-35, 2007] and Banerjee ["Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data," J. Mach. Learn. Res., vol. 9, pp. 485-516, Mar. 2008], to estimate covariances having Kronecker product form. It also generalizes the unpenalized ML flip-flop (FF) algorithm of Dutilleul ["The MLE algorithm for the matrix normal distribution," J. Statist. Comput. Simul., vol. 64, pp. 105-123, 1999] and Werner ["On estimation of covariance matrices with Kronecker product structure," IEEE Trans. Signal Process., vol. 56, no. 2, pp. 478-491, Feb. 2008] to estimation of sparse Kronecker factors. We establish that the KGlasso iterates converge pointwise to a local maximum of the penalized likelihood function. We derive high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. Our results establish that KGlasso has significantly faster asymptotic convergence than Glasso and FF. Simulations are presented that validate the results of our analysis.

This paper presents a novel algorithm, based upon the dependent Dirichlet process mixture model (DDPMM), for clustering batch-sequential data containing an unknown number of evolving clusters. The algorithm is derived via a low-variance asymptotic analysis of the Gibbs sampling algorithm for the DDPMM, and provides a hard clustering with convergence guarantees similar to those of the k-means algorithm. Empirical results from a synthetic test with moving Gaussian clusters and a test with real ADS-B aircraft trajectory data demonstrate that the algorithm requires orders of magnitude less computational time than contemporary probabilistic and hard clustering algorithms, while providing higher accuracy on the examined datasets.

Dropout and other feature noising schemes control overfitting by artificially corrupting the training data. For generalized linear models, dropout performs a form of adaptive regularization. Using this viewpoint, we show that the dropout regularizer is first-order equivalent to an L2 regularizer applied after scaling the features by an estimate of the inverse diagonal Fisher information matrix. We also establish a connection to AdaGrad, an online learning algorithm, and find that a close relative of AdaGrad operates by repeatedly solving linear dropout-regularized problems. By casting dropout as regularization, we develop a natural semi-supervised algorithm that uses unlabeled data to create a better adaptive regularizer. We apply this idea to document classification tasks, and show that it consistently boosts the performance of dropout training, improving on state-of-the-art results on the IMDB reviews dataset.

The fundamental aim of clustering algorithms is to partition data points. We consider tasks where the discovered partition is allowed to vary with some covariate such as space or time. One approach would be to use fragmentation-coagulation processes, but these, being Markov processes, are restricted to linear or tree structured covariate spaces. We define a partition-valued process on an arbitrary covariate space using Gaussian processes. We use the process to construct a multitask clustering model which partitions datapoints in a similar way across multiple data sources, and a time series model of network data which allows cluster assignments to vary over time. We describe sampling algorithms for inference and apply our method to defining cancer subtypes based on different types of cellular characteristics, finding regulatory modules from gene expression data from multiple human populations, and discovering time varying community structure in a social network.

We develop a novel method for detection of signals and reconstruction of images in the presence of random noise. The method uses results from percolation theory. We specifically address the problem of detection of multiple objects of unknown shapes in the case of nonparametric noise. The noise density is unknown and can be heavy-tailed. The objects of interest have unknown varying intensities. No boundary shape constraints are imposed on the objects, only a set of weak bulk conditions is required. We view the object detection problem as a multiple hypothesis testing for discrete statistical inverse problems. We present an algorithm that allows to detect greyscale objects of various shapes in noisy images. We prove results on consistency and algorithmic complexity of our procedures. Applications to cryo-electron microscopy are presented.