Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations

Authors' Response

“…We emphasize that the asymptotic results depend not only on the magnitude of p and n i s but also on the assumption that tr 4 i /p 2 → 0 as p → in (A-iv). Note that one can verify whether (A-iv) is true or not very easily by using the cross-data-matrix methodology (Yata and Aoshima, 2010a,b) or the noise-reduction methodology (Yata and Aoshima, 2012). See Remark 1.1 of Section 1 in our original article for the verification of assumptions.…”

A High-Dimensional Two-Sample Test for Non-Gaussian Data under a Strongly Spiked Eigenvalue Model

“…One of the approaches to HDLSS data is studying its geometric representations. Hall et al (2005), Ahn et al (2007) and Yata and Aoshima (2012) gave several noticeable geometric representations of HDLSS data. Also, Jung and Marron (2009) and Jung et al (2012) extended the result of Hall et al (2005).…”

“…As for (a), the noise space has a geometric representation when p is large. Yata and Aoshima (2012) evaluated the amount of the noise by using the geometric representation and gave the noise-reduction (NR) methodology. Ishii et al (2014Ishii et al ( , 2016 derived several asymptotic properties of the first eigenspace and constructed an equality test of covariance matrices by using the NR method when p → ∞ while n is fixed.…”

Overview of object oriented data analysis

“…The modern theory of HDLSS asymptotics was started by Hall et al (2005), who first pointed out some perhaps surprising properties of randomness in high dimensions. Since then this field has been rapidly developing, see for example Yata and Aoshima (2012) and Shen et al (2012bShen et al ( , 2013. Many more open mathematical statistical problems are available in the area of data lying in manifolds.…”