Geometric Ergodicity of Gibbs Samplers in Bayesian Penalized Regression Models

We consider three Bayesian penalized regression models and show that the respective deterministic scan Gibbs samplers are geometrically ergodic regardless of the dimension of the regression problem. We prove geometric ergodicity of the Gibbs samplers for the Bayesian fused lasso, the Bayesian group lasso, and the Bayesian sparse group lasso. Geometric ergodicity along with a moment condition results in the existence of a Markov chain central limit theorem for Monte Carlo averages and ensures reliable output analysis. Our results of geometric ergodicity allow us to also provide default starting values for the Gibbs samplers

High-Dimensional Screening Using Multiple Grouping of Variables

Screening is the problem of finding a superset of the set of non-zero entries in an unknown p-dimensional vector \beta* given n noisy observations. Naturally, we want this superset to be as small as possible. We propose a novel framework for screening, which we refer to as Multiple Grouping (MuG), that groups variables, performs variable selection over the groups, and repeats this process multiple number of times to estimate a sequence of sets that contains the non-zero entries in \beta*. Screening is done by taking an intersection of all these estimated sets. The MuG framework can be used in conjunction with any group based variable selection algorithm. In the high-dimensional setting, where p >> n, we show that when MuG is used with the group Lasso estimator, screening can be consistently performed without using any tuning parameter. Our numerical simulations clearly show the merits of using the MuG framework in practice.Comment: This paper will appear in the IEEE Transactions on Signal Processing. See http://www.ima.umn.edu/~dvats/MuGScreening.html for more detail

Revisiting the Gelman-Rubin Diagnostic

Gelman and Rubin's (1992) convergence diagnostic is one of the most popular methods for terminating a Markov chain Monte Carlo (MCMC) sampler. Since the seminal paper, researchers have developed sophisticated methods for estimating variance of Monte Carlo averages. We show that these estimators find immediate use in the Gelman-Rubin statistic, a connection not previously established in the literature. We incorporate these estimators to upgrade both the univariate and multivariate Gelman-Rubin statistics, leading to improved stability in MCMC termination time. An immediate advantage is that our new Gelman-Rubin statistic can be calculated for a single chain. In addition, we establish a one-to-one relationship between the Gelman-Rubin statistic and effective sample size. Leveraging this relationship, we develop a principled termination criterion for the Gelman-Rubin statistic. Finally, we demonstrate the utility of our improved diagnostic via examples

On-the-fly Historical Handwritten Text Annotation

The performance of information retrieval algorithms depends upon the availability of ground truth labels annotated by experts. This is an important prerequisite, and difficulties arise when the annotated ground truth labels are incorrect or incomplete due to high levels of degradation. To address this problem, this paper presents a simple method to perform on-the-fly annotation of degraded historical handwritten text in ancient manuscripts. The proposed method aims at quick generation of ground truth and correction of inaccurate annotations such that the bounding box perfectly encapsulates the word, and contains no added noise from the background or surroundings. This method will potentially be of help to historians and researchers in generating and correcting word labels in a document dynamically. The effectiveness of the annotation method is empirically evaluated on an archival manuscript collection from well-known publicly available datasets

Learning Surrogate Models of Document Image Quality Metrics for Automated Document Image Processing

Computation of document image quality metrics often depends upon the availability of a ground truth image corresponding to the document. This limits the applicability of quality metrics in applications such as hyperparameter optimization of image processing algorithms that operate on-the-fly on unseen documents. This work proposes the use of surrogate models to learn the behavior of a given document quality metric on existing datasets where ground truth images are available. The trained surrogate model can later be used to predict the metric value on previously unseen document images without requiring access to ground truth images. The surrogate model is empirically evaluated on the Document Image Binarization Competition (DIBCO) and the Handwritten Document Image Binarization Competition (H-DIBCO) datasets

Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields

We present \emph{telescoping} recursive representations for both continuous and discrete indexed noncausal Gauss-Markov random fields. Our recursions start at the boundary (a hypersurface in $\R^d$, $d \ge 1$) and telescope inwards. For example, for images, the telescoping representation reduce recursions from $d = 2$ to $d = 1$, i.e., to recursions on a single dimension. Under appropriate conditions, the recursions for the random field are linear stochastic differential/difference equations driven by white noise, for which we derive recursive estimation algorithms, that extend standard algorithms, like the Kalman-Bucy filter and the Rauch-Tung-Striebel smoother, to noncausal Markov random fields.Comment: To appear in the Transactions on Information Theor