Sparse Principal Component Analysis (sPCA) is a cardinal technique for obtaining combinations of features, or principal components (PCs), that explain the variance of high-dimensional datasets in an interpretable manner. This involves solving a sparsity and orthogonality constrained convex maximization problem, which is extremely computationally challenging. Most existing works address sparse PCA via methods-such as iteratively computing one sparse PC and deflating the covariance matrix-that do not guarantee the orthogonality, let alone the optimality, of the resulting solution when we seek multiple mutually orthogonal PCs. We challenge this status by reformulating the orthogonality conditions as rank constraints and optimizing over the sparsity and rank constraints simultaneously. We design tight semidefinite relaxations to supply high-quality upper bounds, which we strengthen via additional second-order cone inequalities when each PC's individual sparsity is specified. Further, we derive a combinatorial upper bound on the maximum amount of variance explained as a function of the support. We exploit these relaxations and bounds to propose exact methods and rounding mechanisms that, together, obtain solutions with a bound gap on the order of 0%-15% for real-world datasets with p = 100s or 1000s of features and r \in {2, 3} components. Numerically, our algorithms match (and sometimes surpass) the best performing methods in terms of fraction of variance explained and systematically return PCs that are sparse and orthogonal. In contrast, we find that existing methods like deflation return solutions that violate the orthogonality constraints, even when the data is generated according to sparse orthogonal PCs. Altogether, our approach solves sparse PCA problems with multiple components to certifiable (near) optimality in a practically tractable fashion.

翻译：稀疏主成分分析（sPCA）是一种关键技术，用于获取能够以可解释方式解释高维数据集方差的特征组合或主成分（PC）。这涉及求解一个受稀疏性和正交性约束的凸最大化问题，该问题在计算上极具挑战性。现有研究大多通过迭代计算单个稀疏主成分并对协方差矩阵进行降维等方法处理稀疏主成分分析，但这些方法在寻求多个相互正交的主成分时，无法保证所得解的正交性，更无法保证其最优性。我们通过将正交条件重新表述为秩约束，并同时优化稀疏性和秩约束来挑战这一现状。我们设计了紧的半定松弛以提供高质量的上界，并在指定每个主成分独立稀疏度时，通过附加二阶锥不等式进一步强化该上界。此外，我们推导了作为支撑集函数的方差解释最大值的组合上界。利用这些松弛和上界，我们提出了精确方法和舍入机制，两者相结合，在具有p=数百或数千个特征以及r∈{2,3}个分量的真实数据集中，可获得边界间隙在0%-15%之间的解。数值实验表明，我们的算法在方差解释比例方面与表现最佳的方法相当（有时甚至超越），并能系统性地返回稀疏且正交的主成分。相比之下，我们发现降维等现有方法返回的解会违反正交约束，即使数据是根据稀疏正交主成分生成的。总之，我们的方法能以实际可处理的方式，将多分量稀疏主成分分析问题求解到可证明的（接近）最优解。