Lasso and other regularization procedures are attractive methods for variable selection, subject to a proper choice of shrinkage parameter. Given a set of potential subsets produced by a regularization algorithm, a consistent model selection criterion is proposed to select the best one among this preselected set. The approach leads to a fast and efficient procedure for variable selection, especially in high-dimensional settings. Model selection consistency of the suggested criterion is proven when the number of covariates d is fixed. Simulation studies suggest that the criterion still enjoys model selection consistency when d is much larger than the sample size. The simulations also show that our approach for variable selection works surprisingly well in comparison with existing competitors. The method is also applied to a real data set.

Preprint 1 The Lasso under Heteroscedasticity Jinzhu Jia 1, Karl Rohe 1 and Bin Yu 1, 2 Department of Statistics 1 and Department of EECS 2 University of California, Berkeley Abstract: The performance of the Lasso is well understood under the assumptions of the standard linear model with homoscedastic noise. However, in several applications, the standard model does not describe the important features of the data. This paper examines how the Lasso performs on a nonstandard model that is motivated by medical imaging applications. Like all heteroscedas-tic models, the noise terms in this Poisson-like model are not independent of the design matrix. More specifically, this paper studies the sign consistency of the Lasso under a sparse Poisson-like model. In addition to studying sufficient conditions for the sign consistency of the Lasso estimate, this paper also gives necessary conditions for sign consistency. Both sets of conditions are comparable to results for the homoscedastic model, showing that when a measure of the signal to noise ratio is large, the Lasso performs well on both Poisson-like data and homoscedastic data. Simulations reveal that the Lasso performs equally well in terms of model selection performance on both Poisson-like data and homoscedastic data (with properly scaled noise variance), across a range of parameterizations. Taken as a whole, these results suggest that the Lasso is robust to the Poisson-like heteroscedastic noise. Key words and phrases: Lasso, Poisson-like Model, Sign Consistency, Heteroscedas-ticity 1 Introduction The Lasso (Tibshirani, 1996) is widely used in high dimensional regression for variable selection. Its model selection performance has been well studied under a standard sparse and homoskedastic regression model. Several researchers have shown that under sparsity and regularity conditions, the Lasso can select the true model asymptotically even whenp n (Donoho et al., 2006; Meinshausen arXiv:1011.1026v1 To define the Lasso estimate, suppose the observed data are independent pairs { (x i,Y i)} R p R for i 1, 2,...,n following the linear regression model Y i x T i β i, (1) where x T i is a row vector representing the predictors for thei th observation,Y i is the correspondingi th response variable, i's are independent and mean zero noise terms, andβ R p . Let Y (Y 1,...,Y n)T and ( 1, 2,..., n)T R n . The Lasso estimate (Tibshirani, 1996) is then defined as the solution to a penalized least squares problem (with regularization parameterλ): ˆ β (λ) arg min β 1 2 n ‖Y X β‖ 2 2 λ‖β‖ 1, (2) where for some vectorx R k,‖ x ‖ r ( k i 1 x i r) 1/r .

Traditional spectral clustering methods cannot naturally learn the number of communities in a network and often fail to detect smaller community structure in dense networks because they are based upon external community connectivity properties such as graph cuts. We propose an algorithm for detecting community structure in networks called the leader-follower algorithm which is based upon the natural internal structure expected of communities in social networks. The algorithm uses the notion of network centrality in a novel manner to differentiate leaders (nodes which connect different communities) from loyal followers (nodes which only have neighbors within a single community). Using this approach, it is able to naturally learn the communities from the network structure and does not require the number of communities as an input, in contrast to other common methods such as spectral clustering. We prove that it will detect all of the communities exactly for any network possessing communities with the natural internal structure expected in social networks. More importantly, we demonstrate the effectiveness of the leader-follower algorithm in the context of various real networks ranging from social networks such as Facebook to biological networks such as an fMRI based human brain network. We find that the leader-follower algorithm finds the relevant community structure in these networks without knowing the number of communities beforehand. Also, because the leader-follower algorithm detects communities using their internal structure, we find that it can resolve a finer community structure in dense networks than common spectral clustering methods based on external community structure.

The aim of this study is to show the importance of two classification techniques, viz. decision tree and clustering, in prediction of learning disabilities (LD) of school-age children. LDs affect about 10 percent of all children enrolled in schools. The problems of children with specific learning disabilities have been a cause of concern to parents and teachers for some time. Decision trees and clustering are powerful and popular tools used for classification and prediction in Data mining. Different rules extracted from the decision tree are used for prediction of learning disabilities. Clustering is the assignment of a set of observations into subsets, called clusters, which are useful in finding the different signs and symptoms (attributes) present in the LD affected child. In this paper, J48 algorithm is used for constructing the decision tree and K-means algorithm is used for creating the clusters. By applying these classification techniques, LD in any child can be identified.

The CUR decomposition provides an approximation of a matrix $X$ that has low reconstruction error and that is sparse in the sense that the resulting approximation lies in the span of only a few columns of $X$. In this regard, it appears to be similar to many sparse PCA methods. However, CUR takes a randomized algorithmic approach, whereas most sparse PCA methods are framed as convex optimization problems. In this paper, we try to understand CUR from a sparse optimization viewpoint. We show that CUR is implicitly optimizing a sparse regression objective and, furthermore, cannot be directly cast as a sparse PCA method. We also observe that the sparsity attained by CUR possesses an interesting structure, which leads us to formulate a sparse PCA method that achieves a CUR-like sparsity.

We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an ever-increasing demand for methods of dimension reduction in order to undertake the statistical analysis of interest. Our algorithm uses a gradient-based approach which can be used with an arbitrary loss function provided the latter is differentiable. The speed and effectiveness of our algorithm for dimension reduction is demonstrated in the context of supervised classification of some real high-dimensional data sets from the bioinformatics literature.

This technical report is the union of two contributions to the discussion of the Read Paper "Riemann manifold Langevin and Hamiltonian Monte Carlo methods" by B. Calderhead and M. Girolami, presented in front of the Royal Statistical Society on October 13th 2010 and to appear in the Journal of the Royal Statistical Society Series B. The first comment establishes a parallel and possible interactions with Adaptive Monte Carlo methods. The second comment exposes a detailed study of Riemannian Manifold Hamiltonian Monte Carlo (RMHMC) for a weakly identifiable model presenting a strong ridge in its geometry.

Gaussian graphical models are of great interest in statistical learning. Because the conditional independencies between different nodes correspond to zero entries in the inverse covariance matrix of the Gaussian distribution, one can learn the structure of the graph by estimating a sparse inverse covariance matrix from sample data, by solving a convex maximum likelihood problem with an $\ell_1$-regularization term. In this paper, we propose a first-order method based on an alternating linearization technique that exploits the problem's special structure; in particular, the subproblems solved in each iteration have closed-form solutions. Moreover, our algorithm obtains an $\epsilon$-optimal solution in $O(1/\epsilon)$ iterations. Numerical experiments on both synthetic and real data from gene association networks show that a practical version of this algorithm outperforms other competitive algorithms.

Learning linear combinations of multiple kernels is an appealing strategy when the right choice of features is unknown. Previous approaches to multiple kernel learning (MKL) promote sparse kernel combinations to support interpretability and scalability. Unfortunately, this 1-norm MKL is rarely observed to outperform trivial baselines in practical applications. To allow for robust kernel mixtures, we generalize MKL to arbitrary norms. We devise new insights on the connection between several existing MKL formulations and develop two efficient interleaved optimization strategies for arbitrary norms, like p-norms with p>1. Empirically, we demonstrate that the interleaved optimization strategies are much faster compared to the commonly used wrapper approaches. A theoretical analysis and an experiment on controlled artificial data experiment sheds light on the appropriateness of sparse, non-sparse and $\ell_\infty$-norm MKL in various scenarios. Empirical applications of p-norm MKL to three real-world problems from computational biology show that non-sparse MKL achieves accuracies that go beyond the state-of-the-art.

A density ratio is defined by the ratio of two probability densities. We study the inference problem of density ratios and apply a semi-parametric density-ratio estimator to the two-sample homogeneity test. In the proposed test procedure, the f-divergence between two probability densities is estimated using a density-ratio estimator. The f-divergence estimator is then exploited for the two-sample homogeneity test. We derive the optimal estimator of f-divergence in the sense of the asymptotic variance, and then investigate the relation between the proposed test procedure and the existing score test based on empirical likelihood estimator. Through numerical studies, we illustrate the adequacy of the asymptotic theory for finite-sample inference.