Interpreting Equivariant Representations

Response to reviewer t9rk

Thank you for your time and your supportive feedback. Below, we aim to answer your questions – please ask again if you have further questions. **1. Are isometric cross-sections rare?** Unfortunately, we do believe that isometric cross sections are rare. However, permutation actions are commonly encountered, and we therefore find that the result warrants inclusion. **2. Can we prove that random projections are near isometric?** We don’t have any such results at present. We cannot expect to be able to prove full isometry: Quotient spaces such as Z/G are not generally Euclidean. Indeed, even for graphs with a permutation action, there is no guarantee that they even have bounded curvature. For such spaces, you cannot possibly obtain an isometric projection onto a linear representation, neither invariant nor otherwise. So isometry is more than you wish to ask for, and we don’t expect to be able to prove near isometry in general. However, for special cases, or locally, it might be feasible. We will do our best to clarify this in the paper. We agree that the two invariant visualization techniques are rather different, but we also find that they are tied together by attempting to solve the same problem – which is not generally well known either.

Responses to reviewer Nrx9

Thank you for your time and for your feedback on our paper. Below, we seek to answer your questions – please feel free to let us know should you have further questions **1. Why can’t we just use U-map?** The U-map [1] considers situations where data measured as vectors (thus residing in a Euclidean space R^n) actually reside on a Riemannian submanifold M of R^n. U-map tries to unfold this manifold M with a projection of the data onto it. Our situation is actually different. The equivariant representation X is Euclidean, but because of the equivariance, each data point has a large number of valid representatives in X, and these could be measured as scrambled all over X – the assumption that these randomly scrambled representatives would reside on a submanifold M is clearly invalid. This is also illustrated by Figure 1 and 2 in our paper. The data points are, however, uniquely represented on the quotient Z – but Z is not generally Euclidean. It is also not generally a manifold at all, and there is thus no guarantee that the data resides on a manifold. To conclude, U-map is not applicable to our case. Notice that you could use U-map on top of the invariant representation thus making U-map invariant as well. **2. Why are invariant representations more interpretable?** Invariant representations increase interpretability because they only carry one single representative of each data point. For the equivariant representation of N generic graphs with n nodes each, you will have (n!)^N! different embeddings, with different corresponding plots, visualizing the exact same representation of the exact same data, with pairwise distances that vary significantly and discretely from plot to plot. As a direct consequence of this, when using a non-invariant dimensionality reduction methods such as for example PCA, we obtain visualizations (see the first row of Figure 2), which are indeed very hard to interpret, and may lead us to wrong conclusions, e.g. that the model did not train properly, or that it did not pick up on any structure in our data. The invariant representation, on the other hand, is unique, and only continuously deforms the distances from the quotient, similar to what we are used to in dimensionality reduction tools. [1] McInnes, L., Healy, J., & Melville, J. (2018). Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426.

Responses to reviewer f5He

Thank you for your time and efforts reading our paper, and for your constructive feedback. Below, we will try to respond to your questions and concerns. **1. Why don’t you just use an invariant mapping when x and g.x are similar?** We don’t quite understand the reviewer’s comment – we don’t study cases where the inputs x and g.x are similar, we study cases where they are different – e.g. permuted graphs or rotated images. In such applications, when your model predicts a structured output (e.g. a reconstructed graph, or an image segmentation) you want your model to be equivariant, not invariant. You are right, however, that invariant models are also applied to this type of data – the difference consists of whether or not the output should be transformed analogously to the input (as when the output is a segmentation or a reconstruction), or whether the output does not change when the input is transformed (e.g. when the output is a class). When building classification networks, the standard approach is to stack equivariant layers and then end the model with an invariant layer thus obtaining a fully invariant model. **2. Isn’t Section 4 a related work section?** Section 4 is indeed a related work section. We positioned this section here to aid the reader, as the mathematical framework introduced in the previous sections is needed to fully explain how our methods relate to previously published work. Thank you for pointing out that this was confusing; we have added a comment to make this more clear – please see the updated paper. **3. Why would you analyze an intermediate equivariant representation?** Latent codes are widely used for downstream tasks, examples of which have been included below. The key observation here, is that not all these methods can be designed to be invariant, and as such the obtained results may be deceiving. In the paper we therefore include a range of very different examples where the non-invariance of the applied methods poses a problem (i.e. when doing PCA, interpolation or naive classification based on the latent codes). As these methods are very different in nature when applied to the latent codes, we use the phrase “analysis of the representation” to refer to any non-invariant method which could naively be applied directly to the equivariant representation. Important examples how equivariant latent representations are used: - Predictive models (classification, regression) where the intermediate representation is learned with unsupervised or self-supervised pretraining. Classically, the VAEs studied in our papers have been popular for this task; at present masked autoencoders are gaining popularity. In all of these, you predict objects with similar structure (and similar symmetries) as the input. Thus, you would naturally require or encourage (through augmentation) the model to be equivariant. We provide such examples in the paper, in Figures 5 (graph property regression) and 6 (image classification). - Generation/discovery of new objects such as molecules, proteins, or other structures, with sought properties – such analysis is often based on VAEs, which would typically be required to be equivariant. This relates to the interpolation experiment shown in Fig. 3, where interpolation would typically be used to encourage discovery of new molecules that combine the properties of the two points being interpolated. **4. Isn’t the interpolation experiment self satisfying?** The equivariant representation X is convex. However, when performing convex interpolation, it might suffer from the problem of potentially running into non-aligned representatives of the interpolated graphs, images, etc. The cross section Z_s does not have this problem because it only contains one representative of every graph, and these are geometrically aligned (because Z_s is isometric to the quotient). The fact that the cross section Z_s is a convex subset of the equivariant representation X (see Proposition 2) makes the interpolation computationally and mathematically very easy. The purpose of the interpolation experiment is to empirically demonstrate that this makes a real (and not just mathematical) difference. In our experience, this is important for conveying a message to the more empirical parts of the community. **5. Starting by contextualizing the paper would help readability** Thank you for pointing this out. We have adapted the introduction to emphasize the use of autoencoders for learning self- and semi-supervised embeddings as a prominent use case. Note that equivariant representations are either enforced or encouraged (via augmentation) in any autoencoder, including the VAEs and masked autoencoders commonly used for semisupervised and self supervised learning. These representations are, e.g., typically used for downstream classification, discovery of novel structures like proteins or molecules, but also for specialized tasks such as counterfactual explanations or graph/image editing.