Translation equivariance and translation action on type-n steerable vectors

[zeroAska]

In the appendix of the paper, "translations are trivial to deal with". However, while we can use Wiger D matrices as O(3) representations acting on the steerable vectors, I'm confused that how to have R^3 translations acting on the steerable feature vectors, if there are multiple types?

I'm aware that in the message passing frameworks, the convolution are built upon relative positions between two nodes x_j - x_i, but shouldn't this be translational invariant, instead of translation equivariant?

[RobDHess]

Hi,

It is true that relative positions are translation invariant. However, since the feature vector travels with the node as the graph is translated, the feature vector itself is equivariant.

Kind regards,

Rob