Convexification of Permutation-Invariant Sets and an Application to Sparse PCA

We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and sign-invariant norm with a cardinality constraint. This gives a nonlinear formulation for the feasible set of sparse principal component analysis (sparse PCA) and an alternative proof of the $K$-support norm. Second, we characterize the convex hull of sets of matrices defined by constraining their singular values. As a consequence, we generalize an earlier result that characterizes the convex hull of rank-constrained matrices whose spectral norm is below a given threshold. Third, we derive convex and concave envelopes of various permutation-invariant nonlinear functions and their level-sets over hypercubes, with congruent bounds on all variables. Finally, we develop new relaxations for the exterior product of sparse vectors. Using these relaxations for sparse PCA, we show that our relaxation closes $98\%$ of the gap left by a classical SDP relaxation for instances where the covariance matrices are of dimension up to $50\times 50$.