Multi-block CCA constructs linear relationships explaining coherent variations across multiple blocks of data. We view the multi-block CCA problem as finding leading generalized eigenvectors and propose to solve it via a proximal gradient descent algorithm with $\ell_1$ constraint for high dimensional data. In particular, we use a decaying sequence of constraints over proximal iterations, and show that the resulting estimate is rate-optimal under suitable assumptions. Although several previous works have demonstrated such optimality for the $\ell_0$ constrained problem using iterative approaches, the same level of theoretical understanding for the $\ell_1$ constrained formulation is still lacking. We also describe an easy-to-implement deflation procedure to estimate multiple eigenvectors sequentially. We compare our proposals to several existing methods whose implementations are available on R CRAN, and the proposed methods show competitive performances in both simulations and a real data example.

翻译：多区共同国别评估构建线性关系,解释不同数据区块的一致差异。我们认为多区共同国别评估问题是发现领先的通用源码问题,并提议通过对高维数据使用$\ell_1美元约束的近似梯度梯度后移算法加以解决。特别是,我们采用比准度迭代的衰变限制序列,并表明在适当假设下得出的估计是最佳的。虽然前几部工程已经证明,使用迭接方法,对受限制的美元问题具有如此最佳性能,但对于 $\ell_0美元限制的配方,仍然缺乏同样的理论理解水平。我们还描述了一种易于执行的通缩缩程序,以便按顺序估算多个梯度。我们比较了我们的提案与一些现有的方法,这些方法在 RCRAN 上可以实施,而拟议的方法在模拟和真实数据实例中都显示了竞争性的表现。