Non-negative matrix factorization with the generalized Kullback-Leibler divergence (NMF) and latent Dirichlet allocation (LDA) are two popular approaches for dimensionality reduction of non-negative data. Here, we show that NMF with $\ell_1$ normalization constraints on the columns of both matrices of the decomposition and a Dirichlet prior on the columns of one matrix is equivalent to LDA. To show this, we demonstrate that explicitly accounting for the scaling ambiguity of NMF by adding $\ell_1$ normalization constraints to the optimization problem allows a joint update of both matrices in the widely used multiplicative updates (MU) algorithm. When both of the matrices are normalized, the joint MU algorithm leads to probabilistic latent semantic analysis (PLSA), which is LDA without a Dirichlet prior. Our approach of deriving joint updates for NMF also reveals that a Lasso penalty on one matrix together with an $\ell_1$ normalization constraint on the other matrix is insufficient to induce any sparsity.

非负矩阵因式分解也称为广义Kullback-Leibler散度(NMF)和隐含狄利克雷分配(LDA)是两种用于非负数据降维的流行方法。在这篇论文中，我们展示了在对分解的两个矩阵的列添加$\ell_1$归一化约束和一个狄利克雷先验的条件下，NMF与LDA等效。我们的方法还揭示了一个矩阵上的Lasso惩罚和另一个矩阵的$\ell_1$归一化约束是不足以引发任何稀疏性的。

非负矩阵分解与潜在狄利克雷分配之间的联系