A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.

In this paper we study the support recovery problem for single index models Y=f(X⊺β,ε), where f is an unknown link function, X∼Np(0,𝕀p) and β is an s-sparse unit vector such that βi∈{±1s√,0}. In particular, we look into the performance of two computationally inexpensive algorithms: (a) the diagonal thresholding sliced inverse regression (DT-SIR) introduced by Lin et al. (2015); and (b) a semi-definite programming (SDP) approach inspired by Amini & Wainwright (2008). When s=O(p1−δ) for some δ>0, we demonstrate that both procedures can succeed in recovering the support of β as long as the rescaled sample size κ=nslog(p−s) is larger than a certain critical threshold. On the other hand, when κ is smaller than a critical value, any algorithm fails to recover the support with probability at least 12 asymptotically. In other words, we demonstrate that both DT-SIR and the SDP approach are optimal (up to a scalar) for recovering the support of β in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.

A bandwidth selection method is proposed for local linear regression. Our approach is to combine the ideas of optimal bandwidth selection of Hall et al.(1991) in kernel density estimation, and use of direct bias and variance in Fan and Gijbels (1995) for local linear regression. We show that the bandwidth selector has an optimal relative rate of convergence of n-1/2 with n the sample size.

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish optimal rates of convergence for estimating the principal subspace which sharp with respect to all the parameters, thus providing a complete character ization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy an application of Fano's lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible. We then introduce an adaptive procedure for estimating the principal sub space which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construc tion is a reduction scheme which reduces the sparse PCA problem to a high dimensional multivariate regression problem. This method is potentially useful for other related

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