  Sparse Principal Component Analysis (PCA) methods are efficient tools to
reduce the dimension (or the number of variables) of complex data. Sparse
principal components (PCs) are easier to interpret than conventional PCs,
because most loadings are zero. We study the asymptotic properties of these
sparse PC directions for scenarios with fixed sample size and increasing
dimension (i.e. High Dimension, Low Sample Size (HDLSS)). Under the previously
studied spike covariance assumption, we show that Sparse PCA remains consistent
under the same large spike condition that was previously established for
conventional PCA. Under a broad range of small spike conditions, we find a
large set of sparsity assumptions where Sparse PCA is consistent, but PCA is
strongly inconsistent. The boundaries of the consistent region are clarified
using an oracle result.
