论网络模型中的方差最大化渐近性与降维的谱方法
On varimax asymptotics in network models and spectral methods for dimensionality reduction
AI总结：
本文基于方差最大化旋转的新认知，结合逐元素矩阵扰动分析建立其与网络分析、谱方法的关联，证明潜空间随机图模型中经varimax变换的节点嵌入向量的渐近多元正态性，巩固了矩阵分解类方法的推断基础。
摘要：
方差最大化（varimax）因子旋转自H·凯泽提出以来，虽在心理学和统计学从业者中广为应用，但一直受到部分理论学家和数理统计学家的质疑与不信任。如今，K·罗和M·曾的研究提供了全新的基础性洞见：当与基于谱的矩阵截断降维方法结合时，方差最大化旋转在特定类别潜变量模型中可被证明能执行统计估计。我们基于对方差最大化旋转的这一新认知，进一步建立了其与网络分析、以及基于逐元素矩阵扰动分析的谱方法之间的关联。具体而言，本文证明了在特定潜空间随机图模型中，代表低维节点嵌入的、经方差最大化变换的欧氏点云中的向量具有渐近多元正态性。我们探讨了相关概念，包括网络稀疏性、数据去噪以及矩阵秩在潜变量参数化中的作用。总体而言，这些处于经典与当代多元分析交汇点的研究发现，巩固了基于矩阵分解技术的方法论和推断流程。数值示例验证了我们的发现并补充了相关讨论。
英文摘要：
Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by H. Kaiser, have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by K. Rohe and M. Zeng provides new, fundamental insight: varimax rotations provably perform statistical estimation in certain classes of latent variable models when paired with spectral-based matrix truncations for dimensionality reduction. We build on this newfound understanding of varimax rotations by developing further connections to network analysis and spectral methods rooted in entrywise matrix perturbation analysis. Concretely, this paper establishes the asymptotic multivariate normality of vectors in varimax-transformed Euclidean point clouds that represent low-dimensional node embeddings in certain latent space random graph models. We address related concepts including network sparsity, data denoising, and the role of matrix rank in latent variable parameterizations. Collectively, these findings, at the confluence of classical and contemporary multivariate analysis, reinforce methodology and inference procedures grounded in matrix factorization-based techniques. Numerical examples illustrate our findings and supplement our discussion.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
