论最小迹因子分析——旧曲新唱
On Minimum Trace Factor Analysis -- An Old Song Sung to a New Tune
AI总结：
本文提出最小迹因子分析的松弛版本，以应对异方差噪声下的稳健降维，提供理论保证并建立与现有方法的联系，数值实验验证其有效性。
摘要：
降维方法，如主成分分析（PCA）和因子分析，是数据科学中许多问题的核心。然而，对于具有显著异方差噪声的数据，寻找稳健的低维近似存在严重且已被充分理解的挑战。本文引入了最小迹因子分析（MTFA）的一种松弛版本，这是一种凸优化方法，其根源可追溯至1940年Ledermann的工作。这种松弛在不过度拟合异方差扰动方面特别有效，并解决了因子分析中常被提及的Heywood案例以及最近发现的现有谱方法的“病态诅咒”。我们提供了所得低秩子空间准确性的理论保证，以及所提算法计算该矩阵的收敛速率。我们发展了与现有方法（包括HeteroPCA、Lasso和Soft-Impute）的若干有趣联系，以填补已经庞大的低秩矩阵估计文献中的一个重要空白。数值实验将我们的结果与最近提出的几种处理异方差噪声的方法进行了基准比较。
英文摘要：
Dimensionality reduction methods, such as principal component analysis (PCA) and factor analysis, are central to many problems in data science. There are, however, serious and well-understood challenges to finding robust low dimensional approximations for data with significant heteroskedastic noise. This paper introduces a relaxed version of Minimum Trace Factor Analysis (MTFA), a convex optimization method with roots dating back to the work of Ledermann in 1940. This relaxation is particularly effective at not overfitting to heteroskedastic perturbations and addresses the commonly cited Heywood cases in factor analysis and the recently identified "curse of ill-conditioning" for existing spectral methods. We provide theoretical guarantees on the accuracy of the resulting low rank subspace and the convergence rate of the proposed algorithm to compute that matrix. We develop a number of interesting connections to existing methods, including HeteroPCA, Lasso, and Soft-Impute, to fill an important gap in the already large literature on low rank matrix estimation. Numerical experiments benchmark our results against several recent proposals for dealing with heteroskedastic noise.
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。
