A convex integer programming approach for optimal sparse PCA

Principal component analysis (PCA) is one of the most widely used dimensionality reduction tools in scientific data analysis. The PCA direction, given by the leading eigenvector of a covariance matrix, is a linear combination of all features with nonzero loadings---this impedes interpretability. Sparse principal component analysis (SPCA) is a framework that enhances interpretability by incorporating an additional sparsity requirement in the feature weights (factor loadings) while finding a direction that explains the maximal variation in the data. However, unlike PCA, the optimization problem associated with the SPCA problem is NP-hard. While many heuristic algorithms based on variants of the power method are used to obtain good solutions, they do not provide certificates of optimality on the solution-quality via associated dual bounds. Dual bounds are available via standard semidefinite programming (SDP) based relaxations, which may not be tight and the SDPs are difficult to scale using off-the-shelf solvers. In this paper, we present a convex integer programming (IP) framework to solve the SPCA problem to near-optimality, with an emphasis on deriving associated dual bounds. We present worst-case results on the quality of the dual bound provided by the convex IP. We empirically observe that the dual bounds are significantly better than worst-case performance, and are superior to the SDP bounds on some real-life instances. Moreover, solving the convex IP model using commercial IP solvers appears to scale much better that solving the SDP-relaxation using commercial solvers. To the best of our knowledge, we obtain the best dual bounds for real and artificial instances for SPCA problems involving covariance matrices of size up to 2000 x 2000.



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