论文标题
二进制矩阵的分解：等级关系，唯一性和布尔分解的模型选择
Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition
论文作者
论文摘要
二元矩阵的应用很多。代表矩阵作为少量潜在向量的混合物通过低级分解通常被视为解释和分析数据的有利方法。在这项工作中，我们使用标准算术（真实和非负）和逻辑操作（布尔值和$ \ Mathbb {z} _2 $）检查了二进制矩阵的因素化。我们检查了不同等级之间的关系，并讨论分解何时唯一。特别是，我们表征当布尔分解$ x = w \ land H $具有唯一的$ W $，唯一的$ h $（对于固定$ W $）时，当$ W $和$ h $都是唯一的时，鉴于等级约束。我们介绍了一种称为BMF $ K $的强大布尔模型选择的方法，并在数值示例上显示BMF $ K $不仅准确地确定了正确数量的布尔潜在功能，而且还准确地重建了预定的因素。
The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}_2$). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$ (for a fixed $W$), and when both $W$ and $H$ are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMF$k$, and show on numerical examples that BMF$k$ not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.
