Speaker
Description
High-dimensional, low-sample-size (HDLSS) data are now common in modern scientific fields, such as genomics and medical imaging, where the number of variables significantly exceeds the number of observations. In this talk, we consider the estimation of linear combinations of high-dimensional mean vectors in multi-sample settings. Such inference is a fundamental component of various multivariate analysis techniques, including multi-sample tests and discriminant analysis. In high-dimensional regimes, classical estimators often suffer from severe noise accumulation, leading to inconsistency. To address this problem, sparse estimation techniques, such as thresholding and $L_1$-penalization, have been extensively studied, and their asymptotic properties and practical effectiveness have been discussed in several contexts. However, the performance of these methods depends heavily on the selection of tuning parameters. Common selection procedures, such as cross-validation or information criteria, can be problematic in HDLSS settings due to their heavy computational cost or the deviations from theoretical assumptions. We find that the key to improving estimation accuracy lies in understanding the geometric relationship between the estimation target and high-dimensional noise. Based on this geometric structure, we propose a novel sparse estimation method for high-dimensional mean vectors. The proposed approach effectively mitigates the parameter selection problem, operates with low computational cost, and achieves consistency under HDLSS conditions, as demonstrated both theoretically and through numerical simulations. Finally, we apply our methodology to high-dimensional discriminant analysis and evaluate its performance using real-world high-dimensional datasets.