# What is the generalization of binary/boolean matrix factorization to fuzzy logics called?

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.