Search

We are interested in the problem of two-sample network hypothesis testing: given two networks with the same set of nodes, we wish to test whether the underlying Bernoulli probability matrices of the two networks are the same or not. We propose Interlacing Balance Measure (IBM) as a new two-sample testing approach. We consider the {\it Degree-Corrected Mixed-Membership (DCMM)} model for undirected networks, where we allow severe degree heterogeneity, mixed-memberships, flexible sparsity levels, and weak signals. In such a broad setting, how to find a test that has a tractable limiting null and optimal testing performances is a challenging problem. We show that IBM is such a test: in a broad DCMM setting with only mild regularity conditions, IBM has $N(0,1)$ as the limiting null and achieves the optimal phase transition. While the above is for undirected networks, IBM is a unified approach and is directly implementable for directed networks. For a broad directed-DCMM (extension of DCMM for directed networks) setting, we show that IBM has $N(0, 1/2)$ as the limiting null and continues to achieve the optimal phase transition. We have also applied IBM to the Enron email network and a gene co-expression network, with interesting results., Comment: 92 pages

The knockoff filter is a recent false discovery rate (FDR) control method for high-dimensional linear models. We point out that knockoff has three key components: ranking algorithm, augmented design, and symmetric statistic, and each component admits multiple choices. By considering various combinations of the three components, we obtain a collection of variants of knockoff. All these variants guarantee finite-sample FDR control, and our goal is to compare their power. We assume a Rare and Weak signal model on regression coefficients and compare the power of different variants of knockoff by deriving explicit formulas of false positive rate and false negative rate. Our results provide new insights on how to improve power when controlling FDR at a targeted level. We also compare the power of knockoff with its propotype - a method that uses the same ranking algorithm but has access to an ideal threshold. The comparison reveals the additional price one pays by finding a data-driven threshold to control FDR., Comment: 67 pages, 13 figures

As a principled dimension reduction technique, factor models have been widely adopted in social science, economics, bioinformatics, and many other fields. However, in high-dimensional settings, conducting a 'correct' Bayesianfactor analysis can be subtle since it requires both a careful prescription of the prior distribution and a suitable computational strategy. In particular, we analyze the issues related to the attempt of being "noninformative" for elements of the factor loading matrix, especially for sparse Bayesian factor models in high dimensions, and propose solutions to them. We show here why adopting the orthogonal factor assumption is appropriate and can result in a consistent posterior inference of the loading matrix conditional on the true idiosyncratic variance and the allocation of nonzero elements in the true loading matrix. We also provide an efficient Gibbs sampler to conduct the full posterior inference based on the prior setup from Rockova and George (2016)and a uniform orthogonal factor assumption on the factor matrix.

The spiked covariance model has gained increasing popularity in high-dimensional data analysis. A fundamental problem is determination of the number of spiked eigenvalues, $K$. For estimation of $K$, most attention has focused on the use of $top$ eigenvalues of sample covariance matrix, and there is little investigation into proper ways of utilizing $bulk$ eigenvalues to estimate $K$. We propose a principled approach to incorporating bulk eigenvalues in the estimation of $K$. Our method imposes a working model on the residual covariance matrix, which is assumed to be a diagonal matrix whose entries are drawn from a gamma distribution. Under this model, the bulk eigenvalues are asymptotically close to the quantiles of a fixed parametric distribution. This motivates us to propose a two-step method: the first step uses bulk eigenvalues to estimate parameters of this distribution, and the second step leverages these parameters to assist the estimation of $K$. The resulting estimator $\hat{K}$ aggregates information in a large number of bulk eigenvalues. We show the consistency of $\hat{K}$ under a standard spiked covariance model. We also propose a confidence interval estimate for $K$. Our extensive simulation studies show that the proposed method is robust and outperforms the existing methods in a range of scenarios. We apply the proposed method to analysis of a lung cancer microarray data set and the 1000 Genomes data set., Comment: 48 pages, 8 figures, 5 tables