Symmetric nonnegative matrix factorization (symNMF) is a variant of nonnegative matrix factorization (NMF) that allows handling symmetric input matrices and has been shown to be particularly well suited for clustering tasks. In this paper, we present a new model, dubbed off-diagonal symNMF (ODsymNMF), that does not take into account the diagonal entries of the input matrix in the objective function. ODsymNMF has three key advantages compared to symNMF. First, ODsymNMF is theoretically much more sound as there always exists an exact factorization of size at most n(n − 1)/2 where n is the dimension of the input matrix. Second, it makes more sense in practice as diagonal entries of the input matrix typically correspond to the similarity between an item and itself, not bringing much information. Third, it makes the optimization problem much easier to solve. In particular, it will allow us to design an algorithm based on coordinate descent that minimizes the component-wise ℓ₁ norm between the input matrix and its approximation. We prove that this norm is much better suited for binary input matrices often encountered in practice. We also derive a coordinate descent method for the component-wise ℓ₂ norm, and compare the two approaches with symNMF on synthetic and document datasets.

对称非负矩阵分解（symNMF）是非负矩阵分解（NMF）的一种变体，它允许处理对称输入矩阵，并且已被证明特别适合于聚类任务。在本文中，我们提出了一种新模型，称为非对角symNMF（ODsymNMF），该模型未考虑目标函数中输入矩阵的对角项。与symNMF相比，ODsymNMF具有三个关键优势。首先，ODsymNMF从理论上讲声音要多得多，因为始终存在大小最大为n（n -1）/ 2的精确因式分解，其中n是输入矩阵的维数。其次，在实践中它更有意义，因为输入矩阵的对角线条目通常对应于项目与其自身之间的相似度，而不会带来太多信息。第三，它使优化问题更容易解决。特别地，其将允许我们设计了一个基于坐标下降的算法最小化的逐个分量ℓ _1个输入矩阵和它的近似值之间常态。我们证明了该规范更适合于实践中经常遇到的二进制输入矩阵。我们还推导出逐分量的坐标下降法ℓ ₂规范，并与symNMF的合成和文件数据集的两种方法进行比较。