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 index-set, 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.