AtlasD: Automatic Local Symmetry Discovery

The manuscript is missing a few relevant citations

We appreciate the suggestion for additional citations. The mentioned works on approximate equivariance modify equivariant architectures to handle situations where no perfect global symmetry is present. These works are indeed relevant and we will be sure to include them in our revised manuscript.

The fluid simulation benchmark also appears relevant. We have started work towards an experiment, but are currently addressing challenges related to the chaotic and non-local nature of the dataset.

Eq. 3 Why not train the basis using SVD?

SVD achieves orthogonality but is not suitable for encouraging sparsity. Our standard basis regularization explicitly promotes sparsity by penalizing shared non-zero terms among the generators.

“This implies we only need to consider transformations whose determinant has absolute value 1.” Why?

When the component group of a matrix Lie group $G$ is finite, we may find a finite subgroup $H$ that contains at least one element from each connected component [1]. As $H$ is a finite subgroup of $GL(n)$, for each $h\\in H$, $h^{|H|}=I_n$ so $(\\det h)^{|H|}=1$ and $|\\det h|=1$. Hence, each coset has at least one representative whose determinant has absolute value $1$.

Can you make Fig 4 clearer?

Currently, the subfigures are in row-major order. The pink and green heatmap is the computed invariant metric tensor, using the methodology by [2]. We will make the numbering clearer and split this figure into multiple subfigures.

What are the $\Phi_c$ functions in sec. 4.2.1/5.1?

The description you provided is correct. We will be sure to include more details and examples in the main paper.

5.4 Why so few charts?

The chart size was primarily constrained by the fact that when small charts were used, the atmospheric river class would take up the entirety of the chart. So we instead chose to use a smaller number of larger charts.

If a network is trained on a real dataset, it is typically not equivariant

This is a problem with global symmetry discovery when datasets are canonicalized (e.g. rotation of images removed). It is uncommon to perform such canonicalization at the local scale. Thus, our predictor networks will not face this issue and remain equivariant.

4.2.1 Sampling $\eta \sim N(0,I)$ will only enforce stability to small transformations

Although there are many distributions to try, it’s difficult to sample uniformly from the group as the hypothesis space $\mathrm{GL}(n)$ is not compact. To sample more extreme transformations, one can increase the standard deviation. However, due to the unboundedness of the search space, the distribution must necessarily be biased towards the origin.

Why does the discrete discovery algorithm work?

The search space is relatively low-dimensional and is further reduced in dimensionality by our assumption that the component group is finite. Moreover, by setting $K$ to be significantly larger than the expected number of cosets, at least one representative likely converges toward each ground truth coset. This is confirmed experimentally in 5.1 and 5.2.

To verify if our discrete discovery pipeline is able to discover complex groups, we ask it to find the global symmetries of the function $f(x,y)=|x|+|y|$. The ground truth symmetry group is $D_4$ and our algorithm can recover all $8$ elements. https://i.ibb.co/CKy1wvLc/output.png

4.2.2 Even if $C_i C_j^{-1}$ is not in the identity component, they might be redundant

Note that we claim to discover all the cosets, not just the generators of the component group. In practice, once one has all the elements of the component group, it’s easy for a human to identify the generators.

4.2.2 How is the minimization performed? Is the function convex?

The function is not globally convex (especially since in certain directions $\\exp$ is periodic). Nevertheless, we have found gradient descent to be successful empirically and are yet to run into any issues. In case gradient descent fails, one can turn to higher-order methods like L-BFGS.

5.2 What happens if you seed with more than 1 generator?

When we use multiple generators, the algorithm produces one rotational generator and one that corresponds to a weak scale. The rotational generator is still recognizable, though admittedly less accurate than when the algorithm is seeded with 1 generator.

5.3 Assigning a chart to the region of each digit seems unfair

We acknowledge the MNIST experiment has an idealized setting. Its purpose is to demonstrate the viability of the full pipeline and highlight the difference between local and global transformations. In more realistic scenarios (like our PDE or climate examples), one does not know a priori which charts exhibit local symmetries. Despite having less knowledge, our method still succeeds.

References

[1] A note on free subgroups in linear groups, Wang 1981.

[2] Generative Adversarial Symmetry Discovery, Yang et al. 2023.

A major concern with this method is the assumption that a suitable atlas for the dataset is known.

While this assumption may initially appear overly ideal, in practice it is achievable. The primary requirement for an atlas is that the charts are large enough for the function to truly be atlas local, but small enough that each region is approximately flat. In all of our experiments, we only needed minor additional tuning once this relatively weak condition was met.

We showed in section 5.2 that AtlasD is successful under two diverse atlases, implying one has large freedom in the exact atlas they choose. To provide more evidence, we have created an additional atlas for the PDE experiment with heavy overlap, missing regions, and a sheared/noisy chart. https://i.ibb.co/xtgDr7ZF/Figure-1.png

We discover a single generator \\begin{vmatrix} -0.368 & -1.035 \\\\ 1.101 & 0.386 \\end{vmatrix} and both cosets. The noisy chart does worsen the discovered generator, but it remains recognizable as a rotation.

Atlases constructed using data-driven approaches

We believe the referenced works slightly differ from our setup. In particular, these works model an unknown data manifold embedded in higher-dimensional Euclidean spaces. On the other hand, we deal with feature fields on a simple, explicitly known manifold, such as a sphere or 2D region.

Computational complexity w.r.t hyperparameters $k$ and $K$.

$k$ is simple to tune in practice since we are dealing with low-dimensional spaces and only need several reruns during training. Moreover, only $k$ is chosen in the high-to-low process. $K$, the number of cosets, is fixed once.

It would be beneficial to include Figure 15 from the Supplementary Material in the main manuscript.

We agree that Appendix D.4 is important and will be sure to include it and Figure 15 in the main manuscript.

The proposed method identifies the global symmetry group by aggregating multiple local predictor models. Does this approach enhance the robustness of symmetry discovery against noise?

To be specific, we identify the local symmetry group of $\Phi$ by finding the common global symmetries of all $\Phi_c$. We argue that performing symmetry discovery across multiple predictors effectively multiplies the amount of data elements we have available, making our method resilient to noise.

If the local predictors are not well optimized, the discovered symmetry group might be unreliable.

Yes, this is true. However, the predictors generally have easy tasks in our setup, since they only need to predict locally. This gives us an advantage over global predictors as it is inherently more difficult to predict on a global scale. Also, compared to existing works that use a GAN discriminator, our predictor-based approach does not suffer from the training instability issue of GAN.

The authors mention that standard basis regularization provides more interpretable results. What does "interpretable results" mean in this context? Additionally, could you conduct an ablation study to compare the effectiveness of standard basis regularization and cosine regularization?

We provide an ablation in Appendix D.2 to compare standard basis regularization against cosine similarity.

When cosine similarity is used, we are still able to find an orthogonal basis, but each generator has many non-zero elements (Figure 12). This makes it difficult to understand what physical action each generator corresponds to. In contrast, in the basis discovered using standard basis regularization (Figure 4), each generator has only a few non-zero elements which allows it to easily be classified as a boost or rotation. Hence, “interpretability” in this context means sparsity of the generators.

Algorithm 1, though providing a compact summary, is too vague.

We will be sure to include subroutines for each step in Algorithm 1. If there are any other specific points the reviewer feels should be included in the overview itself, we would be happy to add them.

The reasoning for the necessity of local symmetry is not convincing.

The motivation for local symmetry is that arbitrary manifolds (such as a Möbius strip) do not have global symmetries. In such cases, there is nothing for global discovery methods to learn. However, all manifolds do have local symmetries, making them broadly applicable. Such local symmetries are interesting because they can be used as an inductive bias in gauge equivariant neural networks to improve performance in computer vision, climate segmentation, and other real world tasks [1]. On the other hand, the global symmetries that prior works discover are incompatible with gauge equivariant networks.

The experiments focus only on locally varying symmetry. It is also important to demonstrate that the method consistently discovers the same symmetry for every chart when only global symmetry is present, which prior methods can easily achieve.

We clarify that the atlas equivariance group describes the global symmetries of the local predictors, where each local predictor is the task function restricted to a particular chart. Crucially, it is the common symmetry group for these predictors, rather than varying with each chart. This means that in our experiments, we have in fact been discovering the symmetry group that is the same for all charts.

I also wonder about the memory and time complexity required for learning the group generators.

We include an analysis of space and runtime in Appendix E. In summary, the time and space complexity scale linearly with the number of charts.

Additionally, the paper should provide guidelines on how many neighborhoods are needed

The primary requirement for an atlas is that the charts are large enough for the function to truly be atlas local, but small enough that each region is approximately flat. The exact number of charts is a hyperparameter that may depend on domain-specific factors (e.g. geometry, boundary, data coverage). However, Sec 5.2 demonstrates that our algorithm works under a diverse set of atlases, indicating its robustness to the chart size and count in practice.

I do not see how the proposed loss functions or parameterization are specifically designed for discovering local symmetry.

Compared to those in LieGG or LieGAN, the loss function and parameterization used by AtlasD are specifically tailored to local symmetry discovery. LieGG requires one row in the polarization matrix for every single output pixel in the dataset (Appendix D.4). However, local symmetry datasets are detailed feature fields, making this method impractical due to the memory required. On the other hand, an approach based on LieGAN requires adversarial training, which is often unstable. Moreover, a discriminator-based setup cannot be used in the discrete discovery algorithm because losses for different cosets are no longer comparable. Hence, the predictor loss method used by AtlasD is most applicable to discovering the full local symmetry group.

The reviewer is correct that the discrete discovery is somewhat independent of local symmetry, but we argue that this only boosts our contribution as our method can then be applied to broader domains.

References

[1] Gauge equivariant convolutional networks and the icosahedral cnn, Taco Cohen et al. 2019