Given a matrix of boolean values $\mathbf{X} \in \mathbb{B}^{M \times N} = \{\top, \bot\}^{M \times N}$, the binary/boolean matrix factorization (BMF) problem is to find $\mathbf{U} \in \mathbb{B}^{M \times K}$ and $\mathbf{V} \in \mathbb{B}^{K \times N}$ for some fixed $K$ that minimize $\sum_{i, j} d(x_{ij}, \hat{x}_{ij})$, where $\hat{x}_{ij} = \bigvee_k u_{ik} \land v_{kj}$ and $d$ is some boolean metric.

BMF can be generalized to t-norm fuzzy logics (with involutive negation) by replacing $\mathbb{B}$ with the closed unit interval $[0, 1]$, $\land$ with a t-norm $*$, and $\lor$ with the t-conorm that is dual to $*$ assuming the canonical negator $1 - x$. The objective can similarly be restated in terms of minimizing a unit metric, rather than a boolean metric.

I have found it difficult to locate literature on this generalization of BMF because I do not know how it is commonly referred to. Any pointers to theoretical or experimental results on the topic are appreciated.


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