Optimal classification in sparse Gaussian graphic model

Consider a two-class classification problem where the number of features is much larger than the sample size. The features are masked by Gaussian noise with mean zero and covariance matrix $\Sigma$, where the precision matrix $\Omega=\Sigma^{-1}$ is unknown but is presumably sparse. The useful features, also unknown, are sparse and each contributes weakly (i.e., rare and weak) to the classification decision. By obtaining a reasonably good estimate of $\Omega$, we formulate the setting as a linear regression model. We propose a two-stage classification method where we first select features by the method of Innovated Thresholding (IT), and then use the retained features and Fisher's LDA for classification. In this approach, a crucial problem is how to set the threshold of IT. We approach this problem by adapting the recent innovation of Higher Criticism Thresholding (HCT). We find that when useful features are rare and weak, the limiting behavior of HCT is essentially just as good as the limiting behavior of ideal threshold, the threshold one would choose if the underlying distribution of the signals is known (if only). Somewhat surprisingly, when $\Omega$ is sufficiently sparse, its off-diagonal coordinates usually do not have a major influence over the classification decision. Compared to recent work in the case where $\Omega$ is the identity matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current setting is much more general, which needs a new approach and much more sophisticated analysis. One key component of the analysis is the intimate relationship between HCT and Fisher's separation. Another key component is the tight large-deviation bounds for empirical processes for data with unconventional correlation structures, where graph theory on vertex coloring plays an important role.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1163 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fixed Boundary Flows

We consider the fixed boundary flow with canonical interpretability as principal components extended on the non-linear Riemannian manifolds. We aim to find a flow with fixed starting and ending point for multivariate datasets lying on an embedded non-linear Riemannian manifold, differing from the principal flow that starts from the center of the data cloud. Both points are given in advance, using the intrinsic metric on the manifolds. From the perspective of geometry, the fixed boundary flow is defined as an optimal curve that moves in the data cloud. At any point on the flow, it maximizes the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. We call the new flow the fixed boundary flow. The rigorous definition is given by means of an Euler-Lagrange problem, and its solution is reduced to that of a Differential Algebraic Equation (DAE). A high level algorithm is created to numerically compute the fixed boundary. We show that the fixed boundary flow yields a concatenate of three segments, one of which coincides with the usual principal flow when the manifold is reduced to the Euclidean space. We illustrate how the fixed boundary flow can be used and interpreted, and its application in real data