  Constructing model-agnostic group equivariant networks, such as equitune
(Basu et al., 2023b) and its generalizations (Kim et al., 2023), can be
computationally expensive for large product groups. We address this by
providing efficient model-agnostic equivariant designs for two related
problems: one where the network has multiple inputs each with potentially
different groups acting on them, and another where there is a single input but
the group acting on it is a large product group. For the first design, we
initially consider a linear model and characterize the entire equivariant space
that satisfies this constraint. This characterization gives rise to a novel
fusion layer between different channels that satisfies an invariance-symmetry
(IS) constraint, which we call an IS layer. We then extend this design beyond
linear models, similar to equitune, consisting of equivariant and IS layers. We
also show that the IS layer is a universal approximator of invariant-symmetric
functions. Inspired by the first design, we use the notion of the IS property
to design a second efficient model-agnostic equivariant design for large
product groups acting on a single input. For the first design, we provide
experiments on multi-image classification where each view is transformed
independently with transformations such as rotations. We find equivariant
models are robust to such transformations and perform competitively otherwise.
For the second design, we consider three applications: language
compositionality on the SCAN dataset to product groups; fairness in natural
language generation from GPT-2 to address intersectionality; and robust
zero-shot image classification with CLIP. Overall, our methods are simple and
general, competitive with equitune and its variants, while also being
computationally more efficient.
