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Sparse Principal Component Analysis with missing observations

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).