SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 134-164, January 2021.
Nonnegative matrix factorizations are often encountered in data mining applications where they are used to explain datasets by a small number of parts. For many of these applications it is desirable that there exists a unique nonnegative matrix factorization up to trivial modifications given by scalings and permutations. This means that model parameters are uniquely identifiable from the data. Rigidity theory of bar and joint frameworks is a field that studies uniqueness of point configurations given some of the pairwise distances. The goal of this paper is to use ideas from rigidity theory to study uniqueness of nonnegative matrix factorizations in the case when nonnegative rank of a matrix is equal to its rank. We characterize infinitesimally rigid nonnegative factorizations, prove that a nonnegative factorization is infinitesimally rigid if and only if it is locally rigid and a certain matrix achieves its maximal possible Kruskal rank, and show that locally rigid nonnegative factorizations can be extended to globally rigid nonnegative factorizations. These results give so far the strongest necessary condition for the uniqueness of a nonnegative factorization. We also explore connections between rigidity of nonnegative factorizations and boundaries of the set of matrices of fixed nonnegative rank. Finally we extend these results from nonnegative factorizations to completely positive factorizations.

中文翻译：

---

刚性理论非负矩阵分解的唯一性

SIAM 矩阵分析与应用杂志，第 42 卷，第 1 期，第 134-164 页，2021 年 1 月。
在数据挖掘应用程序中经常遇到非负矩阵分解，它们用于通过少量部分来解释数据集。对于这些应用程序中的许多应用程序，希望存在一个独特的非负矩阵分解，直到由缩放和排列给出的微不足道的修改。这意味着模型参数可以从数据中唯一识别出来。杆和关节框架的刚性理论是一个研究给定一些成对距离的点配置的唯一性的领域。本文的目的是利用刚性理论的思想来研究在矩阵的非负秩等于其秩的情况下非负矩阵分解的唯一性。我们刻画了无穷小的刚性非负因式分解，证明非负因式分解是无限刚性的，当且仅当它是局部刚性的并且某个矩阵达到其最大可能的 Kruskal 秩，并证明局部刚性非负因式分解可以扩展到全局刚性非负因式分解。到目前为止，这些结果给出了非负因式分解唯一性的最强必要条件。我们还探讨了非负因式分解的刚性与固定非负秩矩阵集的边界之间的联系。最后，我们将这些结果从非负因式分解扩展到完全正因式分解。到目前为止，这些结果给出了非负因式分解唯一性的最强必要条件。我们还探讨了非负因式分解的刚性与固定非负秩矩阵集的边界之间的联系。最后，我们将这些结果从非负因式分解扩展到完全正因式分解。到目前为止，这些结果给出了非负因式分解唯一性的最强必要条件。我们还探讨了非负因式分解的刚性与固定非负秩矩阵集的边界之间的联系。最后，我们将这些结果从非负因式分解扩展到完全正因式分解。