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Sparse CCA: Statistical and Computational Limits

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I will introduce the problem of sparse canonical correlation analysis (sparse CCA ). The first part of the talk will focus on the statistical side. I will argue that sparse CCA has an intrinsic difference from the well studied sparse PCA problem because of the presence of high-dimensional nuisance parameters, namely, the marginal covariance matrices. A somewhat surprising result we derived shows that the minimax rate of sparse CCA is independent of the structure of the marginal covariance matrices. The second part of the talk will focus on the computational side. A novel algorithm is proposed to achieve the minimax rate adaptively under an extra sample size condition. I will present a computational lower bound argument to show that such condition is necessary for all polynomial-time algorithms under the Planted Clique hypothesis. A novel reduction procedure is constructed in order that the lower bound is faithful to the Gaussianity of the model. A byproduct of this argument also provides a computational lower bound for sparse PCA under Gaussian spiked covariance model.

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