High-dimensional matrix-valued data is common in scientific and engineering studies and its classification is a significant topic in current statistics. In practice, the discriminative signals of the matrix covariates are oftentimes low rank and sparse. Motivated by this, we propose a sparse and reduced-rank matrix linear discriminant analysis called “Sr-LDA” for binary classification of high-dimensional matrix-valued data. Specifically, based on the Bayes' linear discriminant rule, we derive the theoretically optimal discriminative matrix-valued covariates under the matrix normal assumptions, and constructed a convex empirical loss function for the estimation of the optimal discriminative matrix-valued covariates under the$\ell _{1}$-norm and nuclear norm penalties. Finite sample error bounds for parameter estimation and the misclassification rate are established. The superior performance of the proposed Sr-LDA is illustrated via extensive simulation and real data studies with comparison to other state-of-the-art classifiers.

We propose a first-order autoregressive (i.e. AR(1)) model for dynamic network processes in which edges change over time while nodes remain unchanged. The model depicts the dynamic changes explicitly. It also facilitates simple and efficient statistical inference methods including a permutation test for diagnostic checking for the fitted network models. The proposed model can be applied to the network processes with various underlying structures but with independent edges. As an illustration, an AR(1) stochastic block model has been investigated in depth, which characterizes the latent communities by the transition probabilities over time. This leads to a new and more effective spectral clustering algorithm for identifying the latent communities. We have derived a finite sample condition under which the perfect recovery of the community structure can be achieved by the newly defined spectral clustering algorithm. Furthermore the inference for a change point is incorporated into the AR(1) stochastic block model to cater for possible structure changes. We have derived the explicit error rates for the maximum likelihood estimator of the change-point. Application with three real data sets illustrates both relevance and usefulness of the proposed AR(1) models and the associate inference methods.

Estimation of the precision matrix (or inverse covariance matrix) is of great importance in statistical data analysis and machine learning. However, as the number of parameters scales quadratically with the dimension $p$, the computation becomes very challenging when $p$ is large. In this paper, we propose an adaptive sieving reduction algorithm to generate a solution path for the estimation of precision matrices under the $\ell_1$ penalized D-trace loss, with each subproblem being solved by a second-order algorithm. In each iteration of our algorithm, we are able to greatly reduce the number of variables in the problem based on the Karush-Kuhn-Tucker (KKT) conditions and the sparse structure of the estimated precision matrix in the previous iteration. As a result, our algorithm is capable of handling data sets with very high dimensions that may go beyond the capacity of the existing methods. Moreover, for the sub-problem in each iteration, other than solving the primal problem directly, we develop a semismooth Newton augmented Lagrangian algorithm with global linear convergence rate on the dual problem to improve the efficiency. Theoretical properties of our proposed algorithm have been established. In particular, we show that the convergence rate of our algorithm is asymptotically superlinear. The high efficiency and promising performance of our algorithm are illustrated via extensive simulation studies and real data applications, with comparison to several state-of-the-art solvers.

Many network data encountered are two-mode networks. These networks are characterized by having two sets of nodes and links are only made between nodes belonging to different sets. While their two-mode feature triggers interesting interactions, it also increases the risk of privacy exposure, and it is essential to protect sensitive information from being disclosed when releasing these data. In this paper, we introduce a weak notion of edge differential privacy and propose to release the degree sequence of a two-mode network by adding non-negative Laplacian noises that satisfies this privacy definition. Under mild conditions for an exponential-family model for bipartite graphs in which each node is individually parameterized, we establish the consistency and Asymptotic normality of two differential privacy estimators, the first based on moment equations and the second after denoising the noisy sequence. For the latter, we develop an efficient algorithm which produces a readily useful synthetic bipartite graph. Numerical simulations and a real data application are carried out to verify our theoretical results and demonstrate the usefulness of our proposal.

Quadratic discriminant analysis (QDA) is a standard tool for classification due to its simplicity and flexibility. Because the number of its parameters scales quadratically with the number of the variables, QDA is not practical, however, when the dimensionality is relatively large. To address this, we propose a novel procedure named DA-QDA for QDA in analyzing high-dimensional data. Formulated in a simple and coherent framework, DA-QDA aims to directly estimate the key quantities in the Bayes discriminant function including quadratic interactions and a linear index of the variables for classification. Under appropriate sparsity assumptions, we establish consistency results for estimating the interactions and the linear index, and further demonstrate that the misclassification rate of our procedure converges to the optimal Bayes risk, even when the dimensionality is exponentially high with respect to the sample size. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed for finding interactions, which is much faster than its competitor in the literature. The promising performance of DA-QDA is illustrated via extensive simulation studies and the analysis of four real datasets.