Inferring the covariance matrix of multivariate data is of great interest in statistics, finance, and data science. It is often carried out via the maximum likelihood estimation (MLE) principle, which seeks a covariance matrix estimator maximizing the observed data likelihood. However, such estimator is usually poor when number of samples is not sufficiently larger than the number of variables. With the assumption that a covariance matrix can be decomposed into a low-rank matrix and a diagonal matrix, factor analysis (FA) model is a popular dimensionality reduction technique in improving the estimation performance. Recently, more and more evidence shows that the covariance matrix of real-valued data may admit the nonnegative correlation structure, which has attracted a lot of interest in some areas like finance and psychometrics. There does not exist any work estimating the covariance matrix simultaneously satisfying both structures. In this paper, we propose an MLE problem formulation for covariance matrix considering jointly the low-rank FA model and nonnegative correlation structures. Since the proposed problem formulation is an intractable non-convex problem, a block coordinate descent algorithm is further proposed to solve a relaxed version of our proposed formulation. The superior performance of our proposed formulation and the algorithm are verified through numerical simulations on both synthetic data and real market data.

中文翻译：

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非负相关低秩因子模型下的协方差矩阵估计

推断多元数据的协方差矩阵在统计学、金融和数据科学中引起了极大的兴趣。它通常通过最大似然估计 (MLE) 原理来执行，该原理寻求协方差矩阵估计器来最大化观察到的数据似然性。然而，当样本数量不足以大于变量数量时，这种估计器通常很差。假设协方差矩阵可以分解为低秩矩阵和对角矩阵，因子分析（FA）模型是提高估计性能的一种流行的降维技术。最近，越来越多的证据表明，实值数据的协方差矩阵可能承认非负相关结构，这在金融和心理计量学等领域引起了极大的兴趣。不存在估计同时满足两种结构的协方差矩阵的任何工作。在本文中，我们提出了一种协方差矩阵的 MLE 问题公式，同时考虑了低秩 FA 模型和非负相关结构。由于提出的问题公式是一个棘手的非凸问题，因此进一步提出了块坐标下降算法来解决我们提出的公式的宽松版本。我们提出的公式和算法的优越性能通过对合成数据和真实市场数据的数值模拟得到验证。由于提出的问题公式是一个棘手的非凸问题，因此进一步提出了块坐标下降算法来解决我们提出的公式的宽松版本。我们提出的公式和算法的优越性能通过对合成数据和真实市场数据的数值模拟得到验证。由于提出的问题公式是一个棘手的非凸问题，因此进一步提出了块坐标下降算法来解决我们提出的公式的宽松版本。我们提出的公式和算法的优越性能通过对合成数据和真实市场数据的数值模拟得到验证。