The nonnegative rank of a nonnegative matrix $X$ is the smallest number of nonnegative rank-one factors that sum to $X$. Since computing the nonnegative rank is NP-hard, it is common to circumvent this issue by computing lower and upper bounds. In this paper, we propose non-convex formulations and practical implementations for four important lower bounds for the nonnegative rank, namely the fooling set bound (FSB), the rectangle covering bound (RCB), the hyperplane separation bound (HSB), and the self-scaled bound (SSB). In particular, our algorithm for computing the SSB is the first available in the literature, to the best of our knowledge. It allows us to improve the best known lower bound on the nonnegative rank for some matrices. In some cases, they coincide with the best known upper bound, thereby establishing their exact nonnegative rank for the first time. Moreover, on canonical benchmarks, we show that our non-convex approaches provide a meaningful and often competitive alternative to standard methods. The paper also provides a consolidated reference for the current state of several classical lower bounds on a large number of benchmark matrices.

翻译：非负矩阵 $X$ 的非负秩是将其分解为若干个非负秩一因子之和所需的最少因子个数。由于计算非负秩是NP难问题，通常通过计算其上下界来规避这一难题。本文针对非负秩的四个重要下界——即欺骗集界（FSB）、矩形覆盖界（RCB）、超平面分离界（HSB）和自缩放界（SSB）——提出了非凸优化形式化方法及实用实现方案。其中，据我们所知，本文提出的SSB计算算法是文献中首次实现的。该算法使我们能够改进若干矩阵已知的最佳非负秩下界。在某些情况下，这些下界与已知的最佳上界一致，从而首次确定了这些矩阵的确切非负秩。此外，在标准基准测试中，我们证明了所提出的非凸方法能提供有意义且常具竞争力的替代方案。本文还为大量基准矩阵上若干经典下界的最新研究现状提供了综合性参考。