Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms
Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms
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发表时间：
2022-02
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通讯作者：
Joshua Agterberg;Jeremias Sulam
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作者：
Joshua Agterberg;Jeremias Sulam
稀疏主成分分析 (PCA) 是应用统计学众多子领域的流行工具。虽然几个结果描述了主特征向量的恢复误差，但这些结果通常采用谱或 Frobenius 范数。在本文中，我们为一般高维亚高斯设计下的稀疏 PCA 提供了入门 $\ell_{2,\infty}$ 界限。特别是，我们的结果适用于任何以高概率选择正确支持的算法，即那些稀疏的算法。我们的界限通过提供估计误差的更精细的表征来改进已知结果，并且我们的证明使用最近为条目子空间扰动理论开发的技术。
Sparse Principal Component Analysis (PCA) is a prevalent tool across a plethora of subfields of applied statistics. While several results have characterized the recovery error of the principal eigenvectors, these are typically in spectral or Frobenius norms. In this paper, we provide entrywise $\ell_{2,\infty}$ bounds for Sparse PCA under a general high-dimensional subgaussian design. In particular, our results hold for any algorithm that selects the correct support with high probability, those that are sparsistent. Our bound improves upon known results by providing a finer characterization of the estimation error, and our proof uses techniques recently developed for entrywise subspace perturbation theory.