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.

翻译：SCOTLASS是第一个稀疏主成分分析(SPCA)模型，该模型通过对测量变量施加额外的l1范数约束以获得稀疏载荷。由于在l1球/球面与l2球/球面交集上寻找投影存在困难，早期求解SCOTLASS问题的方法主要聚焦于罚函数法或条件梯度法。本文对SCOTLASS问题进行了重新审视，根据约束集的不同，分别记作SPCA-P1（使用l1球与l2球交集）、SPCA-P2（使用l1球面与l2球面交集）或SPCA-P3（使用l1球与l2球面交集）。我们证明了SPCA-P1与SPCA-P3解的等价性，并表明在大多数情况下SPCA-P2与SPCA-P3的解相同。随后，通过采用l1球/球面与l2球/球面交集上的投影方法，我们为SPCA-P1、SPCA-P2和SPCA-P3问题设计了梯度投影法(简称GPSPCA)和近似牛顿算法(简称ANSPCA)，并证明了所提GPSPCA和ANSPCA算法的全局收敛性。最后，我们在MATLAB环境下进行了多项数值实验，以评估所提GPSPCA和ANSPCA算法的性能。仿真结果证实了SPCA-P1与SPCA-P3解相同、且在大多数情况下SPCA-P2与SPCA-P3解相同的论断，并表明ANSPCA在大规模数据上比GPSPCA更快。此外，与典型的SPCA方法（包括l0约束的GPBB算法、l1约束的BCD-SPCAl1算法、l1惩罚的ConGradU和Gpowerl1算法）相比，GPSPCA和ANSPCA整体表现良好，且可用于大规模计算。