**Response:** We sincerely appreciate your evaluation of our rebuttal materials and the active discussion. Here, we further respond to your raised questions with more targeted explanations.
In terms of downstream application scenarios, it is worth clarifying that the geodesic representation accuracy achieved by our approach is adequate for supporting a rich variety of geometry processing and modeling tasks. Note that the popular classical geodesic algorithms (e.g., *heat method* (HM) and *fast marching method* (FMM)) typically produce results with higher than 1\% errors (sometimes even much higher for anisotropic meshes). Hence, to answer your question about "*applications where a $10^{-2}$ error in the geodesic error would be acceptable*", basically many papers where (original implementations or variations of) HM and FMM are used to produce geodesic information can be taken as examples. As listed in the following, [A, B, C] involve HM-based and FMM-based approaches for geodesic computation, and [D] simply resorts to constructing k-NN graphs followed by the Floyd’s shortest path algorithm, which leads to even higher errors. Also, it can be used for supporting various interactive geometry processing and modeling tasks, such as texture transfer, decal placement, remeshing, smoothing, splines on surfaces, as reported in [E], in which the authors adopted a low-dimensional Euclidean embedding for computing geodesics with approximation error around 5\%.
-- [A] A. Poulenard, et al., "Multi-Directional Geodesic Neural Networks via Equivariant Convolution," in ACM TOG, 2018.
-- [B] B. L. Bhatnagar, et al., "Multi-Garment Net: Learning to Dress 3D People from Images," in Proc. ICCV, 2019.
-- [C] R. Wiersma, et al., "CNNs on Surfaces using Rotation-Equivariant Features," in ACM TOG, 2020.
-- [D] Z. Li, et al., "Geodesic Self-Attention for 3D Point Clouds," in Proc. NeurIPS, 2022.
-- [E] D. Panozzo, et al., "Weighted Averages on Surfaces," in ACM TOG, 2013.
Moreover, we would like to further explain the actual application scenarios of our approach in an overfitting paradigm. To facilitate the intuitive understanding of what we say "interactive applications with rapid and extensive geodesics querying", here we illustrate a very specific case. In the practical industrial scenarios for game/movie/animation, many important 3D asset models such as characters, animals, or other types of objects, need to be repeatedly manipulated, with extensive querying of geodesics at each time. In this case, it can be highly inefficient to apply conventional computational algorithms for many times, and it is even more impractical to store all pre-computed geodesics for every pair of source-target points due to extremely huge memory cost. For this situation, our NeuroGFs can serve as a good choice. Intuitively, once the offline training (i.e., overfitting) process is completed, the neural network model can be viewed as "an attribute" of the target mesh model, just like other attributes like texture. This trained neural network model is quite compact (around 1MB), fast to online query, and can be permanently stored together with the original mesh model as "a compression of its complete geodesics information".
Besides, as for your comment that suggests showing the plot of \#query vs. total training time, we remind that the offline training time cost is not influenced by the number of online queries, because in our implementation we aim to learn complete geodesics of the whole mesh model, regardless of how many queries the users may specify. This setting is reasonable according to the preceding paragraph explaining our suitable application scenarios. More importantly, even if in some cases training time cost is really the most critical consideration factor, one can have multiple choices to deal with the speed-accuracy trade-off:
-- ***(a) Preparing sparser ground-truth pairs.*** As reported in Table S1 of the supplementary material as well as Table R3 of the uploaded PDF file, one can choose to generate sparser ground-truth pairs to significantly reduce the time cost of pre-processing procedures, and the resulting geodesic representation accuracy can still maintain satisfactory.
-- ***(b) Early stopping of the training process.*** As can be found in our rebuttal response to your [Q1], the fast convergence speed of our NeuroGFs allows us to flexibly control the training time cost while maintaining highly competitive performances.
-- ***(c) Adopting generalizable NeuroGF learning frameworks.*** As designed and comprehensively evaluated in our rebuttal (Figure R1 and Table R4 of the uploaded PDF file), one can directly use the generalizable version of our NeuroGF representation framework. After training, the generalizable NeuroGFs can be directly applied to process unseen shapes and categories while achieving satisfactory performances.