Proof of Thm. 2. We want to show M G(hx)= hM G(x) for all x 2X and h 2 G. From the definition of M G in equation 4, we have M G(hx)= 1P Similar to Yarotsky (2022), we first define Ksym = S g2G gK. Note that Ksym is also a compact set and Ksym X . We want to show that M G,equi(gx)= gM G,equi(x). Hence, ( h(gx) 1gx) is invariant to actions of G. The proof for invariance of M G,inv(x) follows similarly. In addition to properties discussed in section 3.3, here we show that equizero models have autoregressive and invertibility properties. These properties have not been used in the main paper, but we believe they could be of use for future work in this area.

Efficient transfer learning algorithms are key to the success of foundation models on diverse downstream tasks even with limited data. Recent works of Basu et al. (2023) and Kaba et al. (2022) propose group averaging (equitune) and optimizationbased methods, respectively, over features from group-transformed inputs to obtain equivariant outputs from non-equivariant neural networks. While Kaba et al. (2022) are only concerned with training from scratch, we find that equitune performs poorly on equivariant zero-shot tasks despite good finetuning results. We hypothesize that this is because pretrained models provide better quality features for certain transformations than others and simply averaging them is deleterious. Hence, we propose λ-equitune that averages the features using importance weights, λs. These weights are learned directly from the data using a small neural network, leading to excellent zero-shot and finetuned results that outperform equitune. Further, we prove that λ-equitune is equivariant and a universal approximator of equivariant functions. Additionally, we show that the method of Kaba et al. (2022) used with appropriate loss functions, which we call equizero, also gives excellent zero-shot and finetuned performance.

Efficient transfer learning algorithms are key to the success of foundation models on diverse downstream tasks even with limited data. Recent works of Basu et al. (2023) and Kaba et al. (2022) propose group averaging (equitune) and optimization-based methods, respectively, over features from group-transformed inputs to obtain equivariant outputs from non-equivariant neural networks. While Kaba et al. (2022) are only concerned with training from scratch, we find that equitune performs poorly on equivariant zero-shot tasks despite good finetuning results. We hypothesize that this is because pretrained models provide better quality features for certain transformations than others and simply averaging them is deleterious. Hence, we propose λ-equitune that averages the features using importance weights, λs. These weights are learned directly from the data using a small neural network, leading to excellent zero-shot and finetuned results that outperform equitune. Further, we prove that λ-equitune is equivariant and a universal approximator of equivariant functions. Additionally, we show that the method of Kaba et al. (2022) used with appropriate loss functions, which we call equizero, also gives excellent zero-shot and finetuned performance.

Efficient transfer learning algorithms are key to the success of foundation models on diverse downstream tasks even with limited data. Recent works of Basu et al. (2023) and Kaba et al. (2022) propose group averaging (equitune) and optimization-based methods, respectively, over features from group-transformed inputs to obtain equivariant outputs from non-equivariant neural networks. While Kaba et al. (2022) are only concerned with training from scratch, we find that equitune performs poorly on equivariant zero-shot tasks despite good finetuning results. We hypothesize that this is because pretrained models provide better quality features for certain transformations than others and simply averaging them is deleterious. Hence, we propose λ-equitune that averages the features using importance weights, λs.

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.