高效求解多层因子模型，支持任意层级结构。
Fitting Multilevel Factor Models
- 基于递归谢尔曼-莫里森-伍德伯里公式，快速计算正定多层低秩矩阵逆。
- 每轮迭代时间与存储复杂度均为线性，适用于大规模数据。
- 开源实现，适合需要多层级建模的统计与机器学习研究者。
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。提出一种新型快速期望最大化算法，专为多层因子模型设计，以最大化观测数据的似然。该方法可处理任意层次结构，并在每轮迭代中保持线性时间与存储复杂度。关键在于提出一种高效计算正定MLR矩阵逆的新技术：证明正定MLR矩阵的逆仍是具有相同因子稀疏性的MLR矩阵，并利用递归谢尔曼-莫里森-伍德伯里恒等式求得逆矩阵的因子。此外，还提出一个算法，可在线性时间和空间内计算扩展矩阵的乔列斯基分解，从而通过舒尔补获得协方差矩阵。本文附带开源包实现所提方法。
原文摘要 · Abstract (English)
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 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 positive definite 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.
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