  SCoTLASS is the first sparse principal component analysis (SPCA) model which
imposes extra l1 norm constraints on the measured variables to obtain sparse
loadings. Due to the the difficulty of finding projections on the intersection
of an l1 ball/sphere and an l2 ball/sphere, early approaches to solving the
SCoTLASS problems were focused on penalty function methods or conditional
gradient methods. In this paper, we re-examine the SCoTLASS problems, denoted
by SPCA-P1, SPCA-P2 or SPCA-P3 when using the intersection of an l1 ball and an
l2 ball, an l1 sphere and an l2 sphere, or an l1 ball and an l2 sphere as
constrained set, respectively. We prove the equivalence of the solutions to
SPCA-P1 and SPCA-P3, and the solutions to SPCA-P2 and SPCA-P3 are the same in
most case. Then by employing the projection method onto the intersection of an
l1 ball/sphere and an l2 ball/sphere, we design a gradient projection method
(GPSPCA for short) and an approximate Newton algorithm (ANSPCA for short) for
SPCA-P1, SPCA-P2 and SPCA-P3 problems, and prove the global convergence of the
proposed GPSPCA and ANSPCA algorithms. Finally, we conduct several numerical
experiments in MATLAB environment to evaluate the performance of our proposed
GPSPCA and ANSPCA algorithms. Simulation results confirm the assertions that
the solutions to SPCA-P1 and SPCA-P3 are the same, and the solutions to SPCA-P2
and SPCA-P3 are the same in most case, and show that ANSPCA is faster than
GPSPCA for large-scale data. Furthermore, GPSPCA and ANSPCA perform well as a
whole comparing with the typical SPCA methods: the l0-constrained GPBB
algorithm, the l1-constrained BCD-SPCAl1 algorithm, the l1-penalized ConGradU
and Gpowerl1 algorithms, and can be used for large-scale computation.
