We consider the problem of identifying the sparse principal component of a rank-deficient matrix. We introduce auxiliary spherical variables and prove that there exists a set of candidate index-sets (that is, sets of indices to the nonzero elements of the vector argument) whose size is polynomially bounded, in terms of rank, and contains the optimal indexset, i.e. the index-set of the nonzero elements of the optimal solution. Finally, we develop an algorithm that computes the optimal sparse principal component in polynomial time for any sparsity degree.
An algorithm that solves sparse PCA for constant rank matrices in polynomial time was introduced in REF .
Sparse principal component of a rank-deficient matrix
2011 IEEE International Symposium on Information Theory Proceedings
2011 IEEE International Symposium on Information Theory Proceedings
Mathematics Computer Science