低秩矩阵的欧氏表示及其统计应用
Euclidean Representation of Low-Rank Matrices and Its Statistical Applications
摘要：
低秩矩阵在统计学、机器学习、信号处理、优化和应用数学中无处不在。在本文中，我们提出了一个新的和用户友好的低秩矩阵的欧几里得表示框架。相应地，我们建立了一个技术和理论工具，用于分析低秩矩阵的内在扰动，其中底层的参考矩阵和扰动矩阵都生活在同一个低秩矩阵流形。我们的分析表明，局部周围的参考矩阵，子空间之间的sine-theta距离相当于两个适当选择的规范正交基之间的欧氏距离，绕过正交Procrustes分析。我们还建立了所提出的欧几里得表示函数的规律性，这具有深刻的统计影响和有意义的几何解释。这些技术手段适用于广泛的统计问题。详细考虑的具体应用包括贝叶斯稀疏尖峰协方差模型与非固有损失，在随机块模型的块概率矩阵可能是退化的有效估计，和最小二乘估计双聚类问题。低秩矩阵的内禀扰动分析和正则性定理可能是独立的兴趣。
英文摘要：
Low-rank matrices are pervasive throughout statistics, machine learning, signal processing, optimization, and applied mathematics. In this paper, we propose a novel and user-friendly Euclidean representation framework for low-rank matrices. Correspondingly, we establish a collection of technical and theoretical tools for analyzing the intrinsic perturbation of low-rank matrices in which the underlying referential matrix and the perturbed matrix both live on the same low-rank matrix manifold. Our analyses show that, locally around the referential matrix, the sine-theta distance between subspaces is equivalent to the Euclidean distance between two appropriately selected orthonormal basis, circumventing the orthogonal Procrustes analysis. We also establish the regularity of the proposed Euclidean representation function, which has a profound statistical impact and a meaningful geometric interpretation. These technical devices are applicable to a broad range of statistical problems. Specific applications considered in detail include Bayesian sparse spiked covariance model with non-intrinsic loss, efficient estimation in stochastic block models where the block probability matrix may be degenerate, and least-squares estimation in biclustering problems. Both the intrinsic perturbation analysis of low-rank matrices and the regularity theorem may be of independent interest.
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