Learning Conjoint Attentions for Graph Neural Nets Supplementary Materials

To prove Theorem 1, we need to consider the two directions of the iff conditions. If we are given h(c1,X1) = h(c2,X2), we are able to prove that the conditions mentioned in the theorem are necessary by showing contradictions occur when they are not satisfied. As Eq. (4) equals Eq. (6), we have: X Obviously, the above equation does not hold as the terms in the summation operator are positive. We may now assume S1 = S2 = S. Eliminating the irrational terms in Eq. (4), we have: X Eq. (9) can be simplified and rewritten as: µ1(x) µ2(x) = However, the RHS of Eq. (10) can be an irrational number. It is obvious that the above equality does not hold as the RHS is an irrational number, while LHS is a rational number.