Hypothetical scenario: I have a pool of millions of items, each of which has two associated values: its weight $x$ and its price $y$. I am training a network that has to predict some value $Z$ based on a collections of $3$ items. In other words,
I am training it through an unsupervised learning technique, where I select random items in a random order for every training sample. I know for a fact that the order of items has no meaning whatsoever, in other words:
$$f(x_1,y_1,x_2,y_2,x_3,y_3) = f(x_2,y_2,x_3,y_3,x_1,y_1)=f(x_3,y_3,x_1,y_1,x_2,y_2)=\ ...$$
Should I in this case share the weights between the first and second layer? So the connections from $x_1$ to the next layer have the same weights as $x_2$ and $x_3$ to the next layer (likewise for $y_1$). Or is there a reason I should not share weights?