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In this article, we study the approximate solutions set $\Lambda_b$ of aninconsistent system of $\max-\min$ fuzzy relational equations $(S): A\Box_{\min}^{\max}x =b$. Using the $L_\infty$ norm, we compute by an explicitanalytical formula the Chebyshev distance $\Delta~=~\inf_{c \in \mathcal{C}}\Vert b -c \Vert$, where $\mathcal{C}$ is the set of second members of theconsistent systems defined with the same matrix $A$. We study the set$\mathcal{C}_b$ of Chebyshev approximations of the second member $b$ i.e.,vectors $c \in \mathcal{C}$ such that $\Vert b -c \Vert = \Delta$, which isassociated to the approximate solutions set $\Lambda_b$ in the following sense:an element of the set $\Lambda_b$ is a solution vector $x^\ast$ of a system $A\Box_{\min}^{\max}x =c$ where $c \in \mathcal{C}_b$. As main results, wedescribe both the structure of the set $\Lambda_b$ and that of the set$\mathcal{C}_b$. We then introduce a paradigm for $\max-\min$ learning weightmatrices that relates input and output data from training data. The learningerror is expressed in terms of the $L_\infty$ norm. We compute by an explicitformula the minimal value of the learning error according to the training data.We give a method to construct weight matrices whose learning error is minimal,that we call approximate weight matrices.  Finally, as an application of our results, we show how to learn approximatelythe rule parameters of a possibilistic rule-based system according to multipletraining data.

Max-min Learning of Approximate Weight Matrices from Fuzzy Data