  Singular value decomposition (SVD) based principal component analysis (PCA)
breaks down in the high-dimensional and limited sample size regime below a
certain critical eigen-SNR that depends on the dimensionality of the system and
the number of samples. Below this critical eigen-SNR, the estimates returned by
the SVD are asymptotically uncorrelated with the latent principal components.
We consider a setting where the left singular vector of the underlying rank one
signal matrix is assumed to be sparse and the right singular vector is assumed
to be equisigned, that is, having either only nonnegative or only nonpositive
entries. We consider six different algorithms for estimating the sparse
principal component based on different statistical criteria and prove that by
exploiting sparsity, we recover consistent estimates in the low eigen-SNR
regime where the SVD fails. Our analysis reveals conditions under which a
coordinate selection scheme based on a \textit{sum-type decision statistic}
outperforms schemes that utilize the $\ell_1$ and $\ell_2$ norm-based
statistics. We derive lower bounds on the size of detectable coordinates of the
principal left singular vector and utilize these lower bounds to derive lower
bounds on the worst-case risk. Finally, we verify our findings with numerical
simulations and illustrate the performance with a video data example, where the
interest is in identifying objects.
