I have a set of vectors $V = (v_1, v_2, \cdots, v_i, \cdots, v_n), v_n \in \mathbb{R}^d$ and a partition annotation $L_i$ for each vector $L_i \in \{1, 2, \cdots, K \}$.
My goal is to partition $V$ only based on linear rules per dimension(intersections of axis-aligned hyperplanes) such that the final output matches the annotation.
For example, $v_1 = (1, 0, 1), v_2 = (2, 2, 1), v_3 = (1, 1, 2), L_1=1, L_2=2, L_3=3$.
Then i can find a rule $(v_{i0} > 2 ) \to 2, (v_{i0} < 2 \& v_{i1} < 1) \to 1, (v_{i0} < 2 \& v_{i1} \geq 1) \to 3$ which matches the annotation.
For a general dataset, how can one determine if there exists a list decision rule that matches the annotation? Also if there don't exist, how can one find a most similar list decision rule such that the annotations are as close as possible?