Given a random vector X, we address the question of linear separability of X, that is, the task of finding a linear operator W such that we have (S1,…,SM)=(WX) with statistically independent random vectors Si. As this requirement alone is already fulfilled trivially by X being independent of...

Principal Components are usually hard to interpret. Sparseness is considered as one way to improve interpretability, and thus a trade-off between variance explained by the components and sparseness is frequently sought. In this note we address the problem of simultaneous maximization of variance...

Modified estimators for the contribution rates of population eigenvalues are given under an elliptically contoured distribution. These estimators decrease the bias of the classical estimator, i.e. the sample contribution rates. The improvement of the modified estimators over the classical...

This paper addresses the problem of reconstructing a low-rank signal matrix observed with additive Gaussian noise. We first establish that, under mild assumptions, one can restrict attention to orthogonally equivariant reconstruction methods, which act only on the singular values of the observed...

In the spiked covariance model for High Dimension Low Sample Size (HDLSS) asymptotics where the dimension tends to infinity while the sample size is fixed, a few largest eigenvalues are assumed to grow as the dimension increases. The rate of growth is crucial as the asymptotic behavior of the...

In this article, we propose a new estimation methodology to deal with PCA for high-dimension, low-sample-size (HDLSS) data. We first show that HDLSS datasets have different geometric representations depending on whether a ρ-mixing-type dependency appears in variables or not. When the...

In High Dimension, Low Sample Size (HDLSS) data situations, where the dimension d is much larger than the sample size n, principal component analysis (PCA) plays an important role in statistical analysis. Under which conditions does the sample PCA well reflect the population covariance...

In this paper we demonstrate that a higher-ranking principal component of the predictor tends to have a stronger correlation with the response in single index models and sufficient dimension reduction. This tendency holds even though the orientation of the predictor is not designed in any way to...