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.

翻译：基于广义Kullback-Leibler散度的非负矩阵分解（NMF）与潜在狄利克雷分配（LDA）是处理非负数据降维的两种主流方法。本文证明：对分解矩阵的列施加$\ell_1$归一化约束，并对其中一个矩阵的列施加狄利克雷先验的NMF，在数学上等价于LDA。为论证该结论，我们通过在优化问题中引入$\ell_1$归一化约束来显式处理NMF的尺度歧义性，这使得广泛使用的乘法更新（MU）算法能够对两个矩阵进行联合更新。当两个矩阵均被归一化时，联合MU算法可推导出概率潜在语义分析（PLSA），即无狄利克雷先验的LDA。我们推导NMF联合更新的方法同时揭示：仅对其中一个矩阵施加Lasso惩罚而对另一矩阵施加$\ell_1$归一化约束，并不能有效诱导稀疏性。