  Principal component analysis (PCA) is one of the most widely used
dimensionality reduction methods in scientific data analysis. In many
applications, for additional interpretability, it is desirable for the factor
loadings to be sparse, that is, we solve PCA with an additional cardinality
(l0) constraint. The resulting optimization problem is called the sparse
principal component analysis (SPCA). One popular approach to achieve sparsity
is to replace the l0 constraint by an l1 constraint. In this paper, we prove
that, independent of the data, the optimal objective function value of the
problem with l0 constraint is within a constant factor of the the optimal
objective function value of the problem with l1 constraint. To the best of our
knowledge, this is the first formal relationship established between the l0 and
the l1 constraint version of the problem.
