Heteroskedastic PCA: Algorithm, optimality, and applications
Heteroskedastic PCA: Algorithm, optimality, and applications
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DOI：
10.1214/21-aos2074
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发表时间：
2018-10
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影响因子：
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通讯作者：
Anru R. Zhang;T. Cai;Yihong Wu
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文献类型：
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作者：
Anru R. Zhang;T. Cai;Yihong Wu
主成分分析（PCA）和奇异值分解（SVD）在统计学、机器学习和应用数学中有着广泛的应用。在污染的噪声级是均匀的情况下，它已经得到了很好的研究。在本文中，我们认为PCA和SVD的存在下的异方差噪声，这自然会出现在一系列的应用。我们介绍了一个一般的框架heteroskedastic PCA，并提出了一种算法称为HeteroPCA，它涉及迭代估算的对角线条目，以消除由于heteroskedasticity的偏见。该方法在广义协方差模型下计算效率高，且可证明是最优的。一个关键的技术步骤是奇异子空间上的确定性鲁棒摄动分析，这可以是独立的利益。该算法的有效性证明了一套应用程序，包括异方差低秩矩阵去噪，泊松PCA，奇异值分解的异方差和不完整的数据。
Principal component analysis (PCA) and singular value decomposition (SVD) are widely used in statistics, machine learning, and applied mathematics. It has been well studied in the case of homoskedastic noise, where the noise levels of the contamination are homogeneous. In this paper, we consider PCA and SVD in the presence of heteroskedastic noise, which arises naturally in a range of applications. We introduce a general framework for heteroskedastic PCA and propose an algorithm called HeteroPCA, which involves iteratively imputing the diagonal entries to remove the bias due to heteroskedasticity. This procedure is computationally efficient and provably optimal under the generalized spiked covariance model. A key technical step is a deterministic robust perturbation analysis on the singular subspace, which can be of independent interest. The effectiveness of the proposed algorithm is demonstrated in a suite of applications, including heteroskedastic low-rank matrix denoising, Poisson PCA, and SVD based on heteroskedastic and incomplete data.