  In this paper, we study the problem of sparse Principal Component Analysis
(PCA) in the high-dimensional setting with missing observations. Our goal is to
estimate the first principal component when we only have access to partial
observations. Existing estimation techniques are usually derived for fully
observed data sets and require a prior knowledge of the sparsity of the first
principal component in order to achieve good statistical guarantees. Our
contributions is threefold. First, we establish the first information-theoretic
lower bound for the sparse PCA problem with missing observations. Second, we
propose a simple procedure that does not require any prior knowledge on the
sparsity of the unknown first principal component or any imputation of the
missing observations, adapts to the unknown sparsity of the first principal
component and achieves the optimal rate of estimation up to a logarithmic
factor. Third, if the covariance matrix of interest admits a sparse first
principal component and is in addition approximately low-rank, then we can
derive a completely data-driven procedure computationally tractable in
high-dimension, adaptive to the unknown sparsity of the first principal
component and statistically optimal (up to a logarithmic factor).
