Abstract We introduce a novel binary matrix factorization (BMF) approach based on a post-nonlinear mixture model. Unlike the existing BMF methods, which are based on the classical matrix product, the proposed mixture model is equivalent to the Boolean matrix factorization model when the entries of the factor matrices are exactly binary. Consequently, our approach yields interpretable results in the case of overlapping sources and more accurate low-rank binary matrix approximations compared to the state-of-the-art. We propose a simple yet efficient algorithm for solving the proposed BMF problem based on multiplicative update rules. In addition, we provide for the first time in the binary data literature, a necessary and sufficient condition for the uniqueness of the Boolean matrix factorization, as well as several other uniqueness results. The interest of this new approach is illustrated in numerical simulation and on real datasets.

中文翻译：

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使用后非线性混合方法对二元矩阵进行布尔分解

摘要 我们介绍了一种基于后非线性混合模型的新型二元矩阵分解 (BMF) 方法。与现有的基于经典矩阵乘积的 BMF 方法不同，当因子矩阵的条目恰好是二进制时，所提出的混合模型等效于布尔矩阵分解模型。因此，与现有技术相比，我们的方法在重叠源和更准确的低秩二进制矩阵近似的情况下产生可解释的结果。我们提出了一种简单而有效的算法，用于基于乘法更新规则解决所提出的 BMF 问题。此外，我们首次在二进制数据文献中提供了布尔矩阵分解唯一性的充分必要条件，以及其他几个唯一性结果。