We introduce a Bayesian perspective for the structured matrix factorization problem. The proposed framework provides a probabilistic interpretation for existing geometric methods based on determinant minimization. We model input data vectors as linear transformations of latent vectors drawn from a distribution uniform over a particular domain reflecting structural assumptions, such as the probability simplex in Nonnegative Matrix Factorization and polytopes in Polytopic Matrix Factorization. We represent the rows of the linear transformation matrix as vectors generated independently from a normal distribution whose covariance matrix is inverse Wishart distributed. We show that the corresponding maximum a posteriori estimation problem boils down to the robust determinant minimization approach for structured matrix factorization, providing insights about parameter selections and potential algorithmic extensions.

翻译：我们提出了一种针对结构化矩阵分解问题的贝叶斯视角。该框架为现有基于行列式最小化的几何方法提供了概率解释。我们将输入数据向量建模为潜在向量的线性变换，其中潜在向量取自某个特定域上的均匀分布，该域反映了结构假设，例如非负矩阵分解中的概率单纯形和多面体矩阵分解中的多面体。我们将线性变换矩阵的行表示为独立生成于正态分布的向量，其协方差矩阵服从逆Wishart分布。我们证明，相应的最大后验估计问题归结为用于结构化矩阵分解的鲁棒行列式最小化方法，从而为参数选择和潜在的算法扩展提供了见解。