A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups

The paper proves that the infinite equivariance constraint ρ2(g)Wρ1(g)−1 = W can be written as (ρ2(g) ⊗ ρ1(g−1)> ) vec(W) = vec(W) and reduced to D infinitesimal and M discrete linear constraints, whose joint nullspace characterizes all equivariant W (Equation (2) and Equation (6) in the paper). Theorem 1 (Appendix B) shows necessity and sufficiency by differentiating at the identity for the Lie algebra generators and enforcing the constraints on the discrete generators, using the decomposition g = exp(∑ αi Ai) Π hki for real Lie groups with finitely many components. The candidate solution exactly mirrors this: it vectorizes the constraint, derives the infinitesimal and discrete constraints, proves sufficiency via an ODE/exponential argument, and concludes that the equivariant space is Null(C) after stacking the blocks. Aside from minor notational differences, the arguments coincide with the paper’s proof and assumptions .