We examine a special case of the multilevel factor model, with covariance given by multilevel low rank (MLR) matrix~\cite{parshakova2023factor}. We develop a novel, fast implementation of the expectation-maximization (EM) algorithm, tailored for multilevel factor models, to maximize the likelihood of the observed data. This method accommodates any hierarchical structure and maintains linear time and storage complexities per iteration. This is achieved through a new efficient technique for computing the inverse of the positive definite MLR matrix. We show that the inverse of an invertible PSD MLR matrix is also an MLR matrix with the same sparsity in factors, and we use the recursive Sherman-Morrison-Woodbury matrix identity to obtain the factors of the inverse. Additionally, we present an algorithm that computes the Cholesky factorization of an expanded matrix with linear time and space complexities, yielding the covariance matrix as its Schur complement. This paper is accompanied by an open-source package that implements the proposed methods.

翻译：我们研究了一种特殊的多层因子模型，其协方差由多层低秩（MLR）矩阵给出。我们开发了一种新颖、快速的期望最大化（EM）算法实现，专为多层因子模型设计，以最大化观测数据的似然。该方法适用于任何层次结构，并在每次迭代中保持线性的时间和存储复杂度。这是通过一种计算正定MLR矩阵逆的新高效技术实现的。我们证明了可逆的半正定MLR矩阵的逆也是一个具有相同因子稀疏性的MLR矩阵，并且我们使用递归的Sherman-Morrison-Woodbury矩阵恒等式来获取逆矩阵的因子。此外，我们提出了一种算法，以线性的时间和空间复杂度计算扩展矩阵的Cholesky分解，从而将协方差矩阵作为其Schur补获得。本文附带了一个开源软件包，实现了所提出的方法。