We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully selected axis‐aligned random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non‐iterative, so it is not vulnerable to a bad choice of initialization. We provide theoretical guarantees under which our principal subspace estimator can attain the minimax optimal rate of convergence in polynomial time. In addition, our theory provides a more refined understanding of the statistical and computational trade‐off in the problem of sparse principal component estimation, revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate. Numerical studies provide further insight into the procedure and confirm its highly competitive finite sample performance.

中文翻译：

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通过轴对齐的随机投影进行稀疏主成分分析

我们引入了一种新的稀疏主成分分析方法，该方法基于从样本协方差矩阵的精心选择的轴对齐随机投影中收集的特征向量信息。与大多数替代方法不同，我们的算法是非迭代的，因此它不容易受到错误的初始化选择的影响。我们提供了理论上的保证，在这些保证下我们的主要子空间估计器可以在多项式时间内达到最小最大最优收敛速度。此外，我们的理论对稀疏主成分估计问题中的统计和计算折衷提供了更精细的理解，揭示了有效样本量与实现最小极大值最优所需的随机投影数量之间的微妙相互作用。率。