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In astronomy, if we denote the dimension of data as $d$ and the number ofsamples as $n$, we often meet a case with $n \ll d$. Traditionally, such asituation is regarded as ill-posed, and there was no choice but to throw awaymost of the information in data dimension to let $d < n$. The data with $n \lld$ is referred to as high-dimensional low sample size (HDLSS). {}To deal withHDLSS problems, a method called high-dimensional statistics has been developedrapidly in the last decade. In this work, we first introduce thehigh-dimensional statistical analysis to the astronomical community. We applytwo representative methods in the high-dimensional statistical analysismethods, the noise-reduction principal component analysis (NRPCA) andregularized principal component analysis (RPCA), to a spectroscopic map of anearby archetype starburst galaxy NGC 253 taken by the Atacama LargeMillimeter/Submillimeter Array (ALMA). The ALMA map is a typical HDLSS dataset.First we analyzed the original data including the Doppler shift due to thesystemic rotation. The high-dimensional PCA could describe the spatialstructure of the rotation precisely. We then applied to the Doppler-shiftcorrected data to analyze more subtle spectral features. The NRPCA and RPCAcould quantify the very complicated characteristics of the ALMA spectra.Particularly, we could extract the information of the global outflow from thecenter of NGC 253. This method can also be applied not only to spectroscopicsurvey data, but also any type of data with small sample size and largedimension.

High Dimensional Statistical Analysis and its Application to ALMA Map of NGC 253