reviews: 
Comments: The authors attempt to make new connections between prior works on convolutions on meshes. They explore a novel parametrization of the convolution operation, and find that it performs worse than the typical parametrization.
Strengths: - I appreciate the attempt of drawing new connections between prior works.
Weaknesses: I am afraid I do not understand the point of this work.

Any linear map from a scalar signal $s: M \to \mathbb R$ on a $n$-manifold, to another scalar signal can be represented by a function $w : M \times M \to \mathbb R$, via the integral $(s \star w)(x) = \int\_M w(x, y)s(y)dy$. Typically, for a convolution-style operator, one does not want the parameters to depend on $x$, so one chooses a frame/gauge $\omega$ on the manifold and arrives at the "intrinsic manifold convolution" of def 1 of the manuscript, with a parameter $t: \mathbb R^n \to \mathbb R$.

Monti et al (2017) [4] recognize that the gauge can not be chosen uniquely, so choose $J$ frames apply a convolution on each of choice of frame, with different parameters each. In other words, they choose a parameter function $t : \mathbb R^n \to \mathbb R^J$.
The alternative approach suggested by [2,3] is to be equivariant to the choice of frame, leading to constraints on the parameters (and typically the use of non-scalar features such as vectors).

However, what's done in theorem 1 in the equation of $(s \star t)\_{\Delta \theta, w}(u)$, is very different from any of the above. Instead of computing the output signal at one point by a single integral (over either the manifold or over the tangent plane), the authors compute a convolution as two integrals over the tangent plane.
Also, they use one parameterization $\mathbb R^n \to \mathbb R$, and another parametrization $\mathbb R^n \times \mathbb R^n \to \mathbb R$.

The authors appear to suggest that this is similar to what Monti et al (2017) [4] does, but this appears to me as very different. As the authors themselves note in theorem 1, this notation is completely redundant and can be reduced to a single integral. In fact, the equation for $(s \star t)\_{\Delta \theta, w}(u)$ appears to do two convolutions, with different parameters, which should indeed reduce to a single convolution. In the rest of sec 3, the authors make the unsurprising observation that if one of the two convolutions contain a Dirac delta, that convolution is an identity operation, and the double convolution reduces to just the second convolution.

So what's the point of analyzing the double convolution, which no one uses? It's very different from what [4] proposes, so how does it bridge a gap between anything?

In section 5, the authors are considering parametrizations of the double convolution different from the Dirac prior (thus the single convolution) and find that they perform worse. Also, as they involve a double integral, I suspect that they are much slower to compute.

In short, the authors didn't make any new connection between prior works, and proposed a different parametrization of the convolution that performed worse. In case I completely misunderstood the work, I look forward to your clarifications and will reconsider my opinion.

Other points:
- The authors cite [1] for the gauge equivariant convolution. This should be a citation to [2] and also [3] in the context of meshes.
- The authors should compare to [3] in their experiments. Those authors found close to 100% correspondence at 0 geodesic distance for the the FAUST experiment, much better than the numbers reported in the present manuscript.
- It's very confusing to use $w$ (w) and $\omega$ (omega) in the same equation, the first referring to weights and the latter to the gauge. Please choose an alternative notation.


Refs:
- [1] Bronstein, Michael M., Joan Bruna, Taco Cohen, and Petar Veličković. 2021. “Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges.” http://arxiv.org/abs/2104.13478.
- [2] Cohen, Taco S., Maurice Weiler, Berkay Kicanaoglu, and Max Welling. 2019. “Gauge Equivariant Convolutional Networks and the Icosahedral CNN.” http://arxiv.org/abs/1902.04615.
- [3] De Haan, P., M. Weiler, T. Cohen, and M. Welling. 2020. “Gauge Equivariant Mesh CNNs: Anisotropic Convolutions on Geometric Graphs.” https://arxiv.org/abs/2003.05425.
- [4] Monti, Federico, Davide Boscaini, Jonathan Masci, Emanuele Rodolà, Jan Svoboda, and Michael M. Bronstein. 2016. “Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs.” http://arxiv.org/abs/1611.08402.
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: 1 poor
Clarity: 2 fair