Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models
Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models
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DOI：
10.1214/17-aos1625
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发表时间：
2018-07
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影响因子：
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通讯作者：
Amelia Perry;Alexander S. Wein;A. Bandeira;Ankur Moitra
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作者：
Amelia Perry;Alexander S. Wein;A. Bandeira;Ankur Moitra
随机矩阵理论的一个中心问题是理解 Johnstone 提出的尖峰随机矩阵模型的特征值，其中将突出的特征向量（或“尖峰”）植入随机矩阵中。这些分布形成了整个科学领域主成分分析 (PCA) 问题的自然统计模型。 Baik、Ben Arous 和 Peche 表明，尖峰 Wishart 系综渐进地表现出急剧的相变：当尖峰强度高于临界阈值时，可以根据顶部特征值检测尖峰的存在，而低于阈值时顶部特征值不提供任何信息。这些结果构成了我们理解 PCA 何时能够在存在噪声的情况下检测低秩信号的基础。然而，在尖峰的结构假设下，并非所有信息都必然包含在频谱中。我们研究尖峰存在测试的统计极限，包括非光谱测试。我们的结果利用了 Le Cam 的邻接概念，包括： i) 对于高斯维格纳系综，我们表明 PCA 实现了尖峰的某些自然先验的最佳检测阈值。 ii) 对于任何非高斯维格纳系综，PCA 的检测效果不是最优的。然而，PCA 的有效变体通过预变换矩阵条目来实现最佳阈值（对于自然先验）。 iii) 对于高斯 Wishart 系综，PCA 阈值对于正尖峰（对于自然先验）是最佳的，但对于负尖峰情况并不总是如此。
A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or "spike") is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and Peche showed that the spiked Wishart ensemble exhibits a sharp phase transition asymptotically: when the spike strength is above a critical threshold, it is possible to detect the presence of a spike based on the top eigenvalue, and below the threshold the top eigenvalue provides no information. Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise. However, under structural assumptions on the spike, not all information is necessarily contained in the spectrum. We study the statistical limits of tests for the presence of a spike, including non-spectral tests. Our results leverage Le Cam's notion of contiguity, and include: i) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for certain natural priors for the spike. ii) For any non-Gaussian Wigner ensemble, PCA is sub-optimal for detection. However, an efficient variant of PCA achieves the optimal threshold (for natural priors) by pre-transforming the matrix entries. iii) For the Gaussian Wishart ensemble, the PCA threshold is optimal for positive spikes (for natural priors) but this is not always the case for negative spikes.