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Given a set $D = \{d_1, d_2, ..., d_N\}$, a set of some subsets of $D$, $D^\ast$ and a set of classes, $C = \{c_1, c_2, ..., c_M\}$, I want to find function, that maps a sequence $({d_i}_1^\ast, {d_i}_2^\ast, ..., {d_i}_L^\ast), {d_i}_j^\ast \in D^\ast$ to $c_k$, provided, that I have example (training) points $\langle({d_i}_1^\ast, {d_i}_2^\ast, ..., {d_i}_L^\ast), c_j\rangle$ and similar arguments should be mapped to the same corresponding function values.

Here is what I mean by similar. Suppose $D = \{1, 2, 3\}$ and $C = \{c_1, c_2\}$, and the arguments of the following three examples for $c_1$:

$\langle(\{1\}, \{1,2\}, \{1\}), c_1\rangle, \langle(\{1\}, \{1,2\}, \{1,3\}), c_1\rangle, \langle(\{2, 3\}, \{2,3\}, \{1\}), c_1\rangle$

are similar, because an intersection of sequences $\{d^\ast\}_{c_1} = \{\{1\}\}$, whereas examples for $c_2$:

$\langle(\{2, 3\}, \{1\}, \{1\}), c_2\rangle, \langle(\{1\}, \{2,3\}, \{2\}), c_2\rangle, \langle(\{2\}, \{2,3\}, \{3\}), c_2\rangle$

the intersection of sequences $\{d^\ast\}_{c_2} = \{\{2, 3\}\}$, that is different and thus dissimilar from $\{d^\ast\}_{c_1}$. So, in general $x_i \rightarrow c_j$ if $j = arg\max_j|\{d^\ast\}_{c_j}\cap x_i|$.

What is a proper name for such problems?

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