reviews: 
Comments: In this paper, the authors relate three special cases of intrinsic convolutions on surfaces to the generic formulation in terms of gauge and template functions.

They further show how different choices of templates can influence the features learnt in conjunction with the weighting function, naming these choices prior as they may encode assumptions made about the surface properties the network can learn.

From the formulation of Monti, they introduce a Dirac distribution prior taken as the zero variance limit of Gaussian functions. They characterise the set of learnable features for the Dirac prior and claim it is more general than others as it is "not limited by an integral".

Finally, the authors conduct experimental evaluations using different choices of priors, taken to be the density functions of several usual probability distributions. In the experimental setting of learning dense point correspondences, they show the Dirac prior outperforms the other choices.
Strengths: The paper relates three milestone formulations of intrinsic convolutions on surfaces to the general theory of convolution in the tangent space with a choice of gauge. The authors introduce a partial order to the set of priors through a notion of what prior is more powerful - i.e., can a prior learn the same features as another.
Weaknesses: The authors focus on a limited body of work in the geometric deep learning literature, namely, three early formulations of intrinsic convolutions. Furthermore, they only cite one reference regarding all the theoretical framework of gauge equivariant CNNs, Bronstein et al., 2021. Combined, these two factors mean the paper ignores important references on gauge equivariant CNNs published before (Cohen et al.) or at the same time (Weiler et al.) the book cited, as well as other related works on convolutions for surfaces and 3D shapes (e.g., Tangent Convolutions for Dense Prediction in 3D in CVPR 2018, MeshCNN, and others).

The mathematical derivations contain typos (the limit of Gaussian functions for decreasing variance should be for n -> 0 not n -> +oo), and the use of the notation $\Delta_{\theta}$ and $\Delta_{\theta, w}$ should be introduced. The authors should introduce the change of integration domain from [0, 1]^n to BR(0) before Theorem 1. The theorem itself is a direct application of Fubini's theorem and as such is not a particularly strong theoretical result of the paper.

As noted by the authors themselves, the Dirac distribution is not a function. This weakens the link with the theoretical results presented before.

The authors claim to "we show that existing definitions for surface convolutions only differ in their prior assumptions about local surface information". It is known in the community, and the authors show it for the formulation of Monti only, who had already shown their formulation encompasses the previous Geodesic CNN and Anisotropic CNN.

Finally, the experimental evaluation yields important questions (detailed below).
Recommendation: 3: reject, not good enough
Reviewer confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness/correctness: 2 fair
Clarity: 3 good