HOPE: Shape Matching Via Aligning Different K-hop Neighbourhoods

Authors: barakeel  fanseu kamhoua
huamin  qu
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-06 11:04:06

Abstract: Accurate and smooth shape matching is very hard to achieve. This is because for accuracy, one needs unique descriptors (signatures) on shapes that distinguish different vertices on a mesh accurately while at the same time being invariant to deformations. However, most existing unique shape descriptors are generally not smooth on the shape and are not noise-robust thus leading to non-smooth matches. On the other hand, for smoothness, one needs descriptors that are smooth and continuous on the shape. However, existing smooth descriptors are generally not unique and as such lose accuracy as they match neighborhoods (for smoothness) rather than exact vertices (for accuracy). In this work, we propose to use different k-hop neighborhoods of vertices as pairwise descriptors for shape matching. We use these descriptors in conjunction with local map distortion (LMD) to refine an initialized map for shape matching. We validate the effectiveness of our pipeline on benchmark datasets such as SCAPE, TOSCA, TOPKIDS, and others.

Main paper: 
 introduction: 
 Shape matching is a very important task and has been increasingly so with the increase in the availability and affordability of 3D scans [31] . It has important applications including but not limited to shape registration [2, 37] , comparison [1, 13] , recognition [6] , and retrieval [50] .Shapes can undergo different types of transformations which we will group as isometric and nonisometric transformations in this work. Nonrigid isometric transformations preserve most of the geometric properties of the shape i.e., after the transformation, its angles, geodesic distances, scale, connectivity, and other geometric properties are mostly preserved. On the other hand, most geometric properties are not preserved under nonrigid non-isometric transformations. Examples of isometric transformations are rotations and translations (though minute scaling may also be included in this category for nonrigid cases). While examples of non-isometric transformations are splitting, scaling, dilating, and others which change the geometric properties of the shapes.Several works have been proposed to address shape matching. However, most of these methods will either aim for smoothness [12, 17, 19, 13, 1, 34, 39, 37, 44, 31, 20] or for accuracy [45] , but not both. Moreover, most of these methods will focus on specific types of deformations undergone by the shape either isometric [12, 19, 13, 1, 34, 39, 37, 44, 31, 20, 49] or non-isometric [14, 22, 52] . Some methods such as 2D-GEM [22] have also tried to address both accuracy and smoothness in shapes undergoing both isometric and non-isometric deformations. However, 2D-GEM is very parameter-dependent, using different user-defined parameters for isometric shapes than for non-isometric shapes which in real life is not practical as one usually does not know whether one is dealing with isometric or non-isometric deformations. Given these obsevations, there is a need for a pipeline that aims for accuracy and smoothness across different settings without significantly changing the pipeline. In this light, we propose HOPE, a k-hop neighborhood-based refinement technique. The rest of the paper is organized into; (a) preliminaries, (b) related work, (c) HOPE, and (d) experiments, (e) limitations and remarks, and (f) conclusion. 
preliminaries: 
 3D shapes are usually represented by their coordinates X ∈ R n×3 and their triangulations (meshes) which can be used to build an adjacency A ∈ R n×n , with A(i, j) = 1 if vertices i and j are connected in the shape and 0 otherwise.Given two three-dimensional shapes M and N with n M and n N vertices respectively. Though our proposed pipeline (HOPE) and some other baselines work with n M ̸ = n N , we will assume n M = n N = n, and use n for simplicity except when specified. The goal of nonrigid shape matching is to find a meaningful correspondence T : M → N , where T is (a) bijective, (b) continuous (smooth) , and (c) similar vertices should be matched to each other [26, 35, 22] . A measure that captures both the smoothness and the accuracy of the map is the geodesic error. The geodesic error measures how far a map T , maps a vertex from its matching position given by the ground truth map T , where a higher map accuracy will correspond to a higher proportion of vertices having geodesic error 0, and a smoother map will correspond to a huge increase in the geodesic error curve as we move slightly away from error 0 (implying that several vertices though not accurate are mapped to their intended neighborhoods).The two main components of shape matching involve: (1) initializing a map, and (2) refining the initialized map.First, the map is initialized either by using landmarks vertices (i.e., vertices with known correspondences) or by using robust descriptors. These descriptors can be based on: (a) the spectrum of the shape laplacian [8, 4, 33] , (b) The face normals, vertex location, and triangulation [45, 41] , geodesic distances [44, 1] or others [14, 46, 15, 16, 11, 43, 23] . In this work, we will assume that one such initialization approach has been utilized and we have an initial map at hand. Second, the initialized map T 0 can then be refined by either using pair-wise descriptors [36, 26, 52, 48, 14, 38, 22] and solving:EQUATIONor by using vertex-wise descriptors [2, 34, 39, 37, 31, 26, 35, 19, 20, 44, 1, 28] and solving:EQUATIONWhere in equation 1, W M (T t , T t-1 ) ∈ (R) n×n is the pair-wise descriptors of vertices for shape M with its rows and columns aligned using the map T t and the previous iterations map T t-1 respectively. W N is the pair-wise descriptor for shape N . Here for conciseness for aligning rows or columns we will use T (while in reality if the row map is T = argmax(P, dim=-1) the column should be T ′ = argmax(P, dim=-2) where P is the permutation matrix such thatPQ M = Q N and PW M P T = W N While in equation 2, Q M (T t , :) ∈ R n×dis the vertex-wise descriptors for shape M with feature dimension d for each vertex and with its rows aligned according to the map T t , f (T t-1 ) is a function (such as functional map [34] ) to transfer the previous iteration's map to the descriptor space, and Q N ∈ R n×d is the vertex-wise descriptors for shape N .For the map T refined by equations 1 or 2 to have a high accuracy (in the ideal case), the pair-wise descriptors W N (i, :) ∈ R 1×n or vertex-wise descriptors Q N (i, :) ∈ R 1×d will need to be: (a) unique for each vertex i on shape N , and (b) identical to its ground truth corresponding pair-wise feature W M ( T (i), :) ∈ R 1×n or vertex-wise feature Q M ( T (i), :) on shape M. Where T is the ground truth map. Moreover, for smoothness, vertices should be mapped such that they remain in their relative neighborhoods before and after the mapping. 
related work: 
 ZoomOut [31] and other related works [19, 13, 1, 34, 39, 37, 26, 35] all aim for smoothness of the map and focus on the settings in which the shapes have undergone an isometric deformation, such Figure 1 : For sample shapes from the TOPKIDS (first row) and TOSCA (second row), this figure shows the second eigenvector of the mesh laplacian (LBO 2), the second eigenvector of the uniform shape laplacian from the triangulation adjacency (uniform shape laplacian 2), the 2-hop and 6-hop neighborhoods of vertex 100, the second SHOT descriptor and the LMD of vertex 2.that geometric shape properties are preserved. These works use vertex-wise descriptors usually based on some truncated (reduced) basis of specific shape properties, generally leading to some loss of information such as uniqueness. For example, consider the d dimensional vector U(i, :) ∈ R d as the vertex-wise descriptors of vertex i, supposing that the matrix U ∈ R n×d contains the first d eigenvectors of the uniform shape laplacian built from A, it can be seen that U(i, :) is the soft cluster assignment of vertex i, and as such most vertices in the same cluster as i will have similar vertex-wise descriptors. This can be seen in figure 1 where the second eigenvector U(:, 2) ∈ R n is shown for example shapes from TOSCA and TOPKIDS. One can see that U(:, 2) ∈ R n is indeed a soft cluster assignment grouping some vertices in the same cluster together. This explains the success in achieving smoothness by some methods that use the d dimensional spectrum of the shape laplacian.Theorem 3.1 Given the shape descriptor U(i, :) ∈ R d as the vertex-wise descriptors of vertex i, supposing that the matrix U ∈ R n×d contains the first d eigenvectors or left singular vectors of some unique shape pairwise descriptor W. Using W for the map refinement via Functional maps helps group nearby clusters together assuming the functional map is perfectly accurate. Proof 3.1 Recall that given a map T t-1 , and two vertex-wise descriptors U M and U N both ∈ R n×d the functional map C ∈ R d×d is obtained as:C t = min C t ||U M (T t-1 , :) -U N C t ||This functional map can then be used to convert functions in the basis of shape M into those of shape N and vice versa. This is then used to refine the map via solving equation 2 where f (T t ) = C t . One can see that U N C t converts the soft clusters (left singular vectors or first eigenvectors) of shape U N into the basis space of the descriptors (soft clusters) of shape U M , and so solving 2 is basically aligning the soft clusters in the same basis space. Thus solving equation 2 is matching aligned soft clusters.In fact, methods such as ZoomOut [31, 20, 34] which iteratively refine the map T by using an increasing number of eigenvectors of the shape laplacian can be said to be refining the map by increasingly aligning more fine-grained clusters of the shapes. Figure 2 shows that though ZoomOut's recovered maps T on the isometric SCAPE [26] dataset are generally smooth (seen by the rapid increase of the geodesic curve), they are generally not accurate due to the non-uniqueness of the descriptors it uses to refine the initialized map (seen by the proportion of vertices at geodesic error 0).Moreover, though geometric properties may be preserved even for some mild non-isometric deformations, they usually are not preserved when the transformation deviates significantly from isometry. For example, consider the shape Laplace Beltrami Operator (LBO) on which many vertex-wise and pair-wise descriptors are built: where V is the diagonal mass matrix with V(i, i) entry along the diagonal being the vertex area of vertex i, and the matrix S being defined as:EQUATIONEQUATIONwhere neigh(i) denotes the neighborhood of i. It can be observed from the way the LBO is defined that a non-isometric deformation which changes angles, areas, or even merges part of the shape (such as gluing hands to face for example) will change the LBO and by implication its spectrum, hence changing the vertex-wise and pair-wise descriptors based on the LBO. Figure 2 shows the poor performance of DIR [49] and ZoomOut [31] (which use the spectrum of the LBO as vertex-wise descriptors) on a pair of sample shapes from the non-isometric dataset TOPKIDS [26] (which contains shapes with topological noise).Other methods aim for accuracy use more unique descriptors. However, these descriptors are generally not smooth since they generally do not capture neighborhoods and as such do not match neighborhoods. Figure 1 shows the first SHOT [45] descriptor and figure 2 show that while it generally provides an accurate map [45] (seen by the proportion of vertices at geodesic error 0) the map is not smooth (as seen by the lack of rapid increase of the geodesic curve). That is why methods aiming for smoothness [31, 37, 12, 17] only use these unique descriptors for initialization and then use the LBO or other smooth basis for refinement.Others have used the 1-hop [14] or 2-hop [52, 22] neighborhoods of vertices as witnesses to improve an initialized map. Though these methods aim for both accuracy (via uniqueness of vertex neighborhoods), and smoothness via using neighborhoods as witnesses to improve the map (matching neighborhoods that agree best), they are nonetheless affected by the fact that 1-hop or 2-hop neighborhoods may not be unique enough and distant vertices may have similar 1-hop and or 2-hop neighborhoods especially in the presence of symmetries in A. For example, consider GRAMPA [14] which initializes the map by using a graph matching kernel in the full graph spectrum and then uses the 1-hop (or 2-hop as in this figure) neighborhood to refine the map. Figure 2 shows GRAMPA [14] performing well on non-isometric shapes from TOPKIDS where there are few symmetries (and other graph properties) which normally cause the neighborhoods of the vertices to be non-unique. However, figure 2 shows that GRAMPA struggles on a pair of shapes from the nearly isometric dataset SCAPE [26] where shapes have symmetries (and other graph properties) that normally cause the neighborhoods of the vertices to be non-unique. To address this challenge, 2D-GEM [22] proposed to use the concept of local map distortion LMD [49, 48] and the spectrum of the uniform shape laplacian. However, though 2D-GEM seems to perform well on non-isometric and isometric shapes (figure 2 ), 2D-GEM does not generalize, but rather significantly changed their model (depending on user-defined parameters) in order to handle isometric cases as discussed in Section 4.4. 
hope: 
 To address both accuracy and smoothness for correlated meshes undergoing isometric and nonisometric deformations we propose HOPE (k-HOP niEghborhood matching). HOPE is based on the iterative refinement of an initialized map by: (1) using the local map distortion (LMD) [49, 48] to identify poorly matched vertices, and (2) using noise robust k-hop neighborhood-based descriptors for refinement of the maps of these poorly matched vertices.This section is organized as: (a) the local map distortion, (b) k-hop pairwise descriptors, and (c) iterative refinement pipeline and algorithm.Let T : M → N be a map between two shapes. The LMD [49, 22] of the map T at the vertex x i is given as follows:EQUATIONwhereB γ (x i ) = {x j ∈ M | d M (x i , x j ) ≤ γ} is the γ-geodesic ball of x i , V Mis the area element of the mesh of shape M, andDE T (x i , x j ) = |d M (x i , x j ) -d N (T (x i ), T (x j ))|/γ represents a pairwise distance distortion of mapping nearby vertices x i and x j to T (x i ) and T (x j ). A smaller value of D γ (T )(x i ) means a better map continuity of T at the vertex x i , in other words, the local distance at the point x i is well preserved. Based on the above definition of LMD, one can check that if T is an isometric map, thenD γ (T )(x i ) = 0, ∀x i ∈ M, γ > 0. Conversely, if D γ (T )(x i ) = 0, ∀x i ∈ Mfor some γ > 0, then T is isometric. We use the LMD to find well-matched pairs (lmks) i.e.,lmks = {(x i , T (x i ))|D γ (T )(x i ) ≤ ϵ},where ϵ is a threshold, We fix ϵ to be the same values for all dataset for all our experiments. We then call the rest of the vertices on shape M / ∈ lmks as non-landmarks N lmks (i.e., poorly mapped vertices).Given an initial map T 0 matching a fraction β of the vertices n correctly and the fraction 1 -β incorrectly, it has been shown that one can refine the map by (a) using the 1-hop [29] neighborhood contained in the graph adjacencies of the triangulations A M and A N , or (b) using k-hop [52] neighborhoods contained in binary matrices A M,k and A N ,k ∈ R n×n indicating whether vertices are connected at a path of length k on the shape mesh adjacencies (note that k = 1 is the adjacency). The refinement is done by solving:EQUATIONthe main assumption being that 2 the meshes M and N are correlated i.e., their adjacencies are assumed to come from the same parent graph, with their correlation ratio being s. Specifically, given a G(n, p) Erdos Regny parent graph and the correlation s, 1 -s is the probability of independently randomly deleting edges from this graph to either obtain the adjacency of M or that of N . For k = 1 and k = 2, rigorous analysis and tight bounds were given for recovering the ground truth map T (for given values of s, p and β) by solving equation 6 (see [51, 30, 52, 29] for exact bounds).Theorem 4.1 Given the k-hop based descriptors A M,k and A N ,k , solving equation 6 is matching vertices whose neighborhoods agree best under T t-1 .Proof 4.1 Notice that:• columns of A M,k andA N ,k are descriptors for the vertices (rows), and these columns are the k-hop neighborhoods of vertices (rows)• T t-1 is used in equation 6 for first aligning these descriptors i.e., A M,k (:, T t-1 ) rearranges the columns of A M,k thus realigning the descriptors of each vertex (the rows),• each entry K(i, j) in the product K = A M,k (:, T t-1 )A N ,k will be the number of vertices that are common in the k-hop neighborhoods of vertices i and j, after the alignment of the k-hop neighborhoods of vertices in M by A M,k (:, T t-1 )Figure 3 : HOPE (a pipeline) for shape matching.• finding T via equation 6 matches the vertices i, and j whose k-hop neighborhood have most vertices in common based on the alignment T t-1 .However, in the presence of symmetries the next 1-hop and 2-hop neighborhoods may not be unique enough as even distant nodes may have the same 1 and or 2-hop connectivity. To address this nonuniqueness, we follow 2D-GEM, and use the LMD to first detect N lmks vertices. Unlike 2D-GEM which used the laplacian spectrum to refine the map T (N lmks) for N lmks vertices, we instead propose to iteratively use different k-hop neighborhoods for refining the map T (N lmks) for N lmks vertices, i.e., iteratively using nodes at different lengths k to refine T (N mlks). We propose this strategy because: (b) first, 2D-GEM's strategy of using the spectrum will suffer from non-uniqueness according to theorm 3.1, and will also not adapt to non-isometric shapes as discussed in Sections 3 and 4.4. (b) Second, according to theorem 4.1 our strategy will ensure that T is consistent across different neighborhoods. We do so for 1 ≤ k ≤ k max where k max is large. This strategy is inspired by the fact that it was shown that large neighborhood statistics are essential in graph matching [51, 30, 52, 29, 10, 32] , and other graph tasks [18, 5, 47, 21, 22, 53] .Here we show the general pipeline for HOPE which consists of the two aforementioned steps, namely:• (a) Map initialization: we use any robust map initializations e.g., the map initialized from SHOT [45] in our experiments• (b) map refinement for t iteration steps consisting of:-Using the LMD [22, 49, 48] to detect the poorly matched N lmks vertices refine the map for the N lmks vertices via enforcing different noise robust k-hop neighborhoods consistency (for 1 ≤ k ≤ k max ) by solving:EQUATION• (c) returning the final map T tIn this section, we briefly introduce 2D-GEM [22] and highlight some key differences between 2D-GEM and HOPE. Like us, 2D-GEM initializes its map via SHOT [45] Figure 4 : Comparison on TOPKIDS showing HOPE using ϵ as in Section 5.5 for its LMD threshold (same as in the isometric cases), against 2D-GEM with ϵ set as in the isometric case (i.e., using the isometric parameters in [22] rather than ϵ = 100 as in the nonisometric case).and then at each iteration t 2D-GEM refined the map by:• (1) updating the map T by solving one iteration of equation 6 using the 2-hop neighborhoods, meaning k = 2 in equation 6,• (2) finding the lmks well-matched pairs, and poorly matched pairs N lmks pairs using the LMD.• (3) using the lmks vertices to update the map of the N lmks vertices in the spectrum by using the GMWM [52] on a cost matrix built from the spectrum of the laplacian of the two shapes (see their paper for more details). They showed similarity between this approach and the functional map [34] approach.The main two differences between 2D-GEM and HOPE are: (a) HOPE completely removes their step (1), and (b) they used the laplacian (LBO) spectrum via their proposed 2D-graph convolution to refine the map for the N lmks in step (3). However, as mentioned in section 3 and shown in figure 1 , using LBO descriptors loses uniqueness since vertices in similar clusters will be grouped together, and in addition, these descriptors are not robust to non-isometric deformations as discussed in Section 3.A consequence of this is that, 2D-GEM [22] needed to deactivate step ( 2) and (3) of their algorithms (the N lmks refinement step) by setting ϵ = 100 as their LMD threshold when dealing with nonisometric shapes. Hence though effective, their algorithm can be seen as two disjoint algorithms that are used for isometric or non-isometric shapes based on the setting of ϵ. But in real life, in some cases, it is not obvious whether the deformation used is isometric or not. On the other hand, we propose refining the N lmks map based on the preservation of different k-hop connectivity which is a constraint that holds for isometric and non-isometric shapes with correlated triangulations [51, 30, 52] .Figure 4 shows a comparison between HOPE and 2D-GEM on a non-isometric pair of shapes from TOPKIDS using the 2D-GEM ϵ parameters proposed for isometric shapes. It can be seen that when we use the isometric parameters of 2D-GEM on non-isometric shapes, 2D-GEM fails to perform well (the same holds when using their non-isometric parameter ϵ = 100 on isometric shapes.See the algorithm for HOPE in Appendix A. Given that for step (1) HOPE follows DIR [49] and [22] in using LMD, checking the LMD takes O(n). The GMWM used to solve step (2) of HOPE takes O(|N lmks| 2 logn). Hence the total time complexity of HOPE for all t iterations is O(t(|N lmks| 2 logn + n)). 
experiments: 
 We report experimental results that validate the effectiveness, efficiency, and generalization ability of HOPE in the matching of nearly-isometric and non-isometric 3D shapes.All experiments are conducted in Matlab 2023 on a Windows 11 system with 32GB RAM and Intel(R) i5 13500 CPU @ 2.50-4.8GHz.We evaluate the performance of HOPE on two nearly isometric benchmark datasets TOSCA [7] ,and SCAPE [3] , as well as on the non-isometric dataset SHREC'16 (TOPKIDS) [26] , TOPKIDS contains 25 shapes of the same class with up to 12K vertices, undergoing near-isometric deformations in addition to large topological noise (such as merging hands to thighs) which results in n M ̸ = n N . TOSCA consists of 80 shapes in 8 different categories (human and animal shapes) with vertex numbers ranging from 4k to 50k. SCAPE has 71 shapes (12,500 vertices for each) of the same person with different poses.Furthermore, to see the generalization abilities of HOPE, we equally used datasets SCAPE_r [9] , FAUST_r [9] , and TOSCA_r [9] , which are remeshed shape datasets. The SCAPE_r consists of the same 71 shapes from the SCAPE dataset, but remeshed, while the FAUST_r (TOSCA_r) likewise contains the same FAUST (TOSCA) datasets but remeshed. We used the same 71 test pairs SCAPE_r as for SCAPE, and for FAUST_r and TOSCA_r we followed [9]We use the geodesic error as our error metric [14, 22] . Given that the map of an algorithm maps x i ∈ M to x j ∈ N , and the true map maps x i to x * j , the geodesic error is defined as e(x i ) =d N (xj ,x * j ) diam(N ), where d N denotes the geodesic distance on N , and diam(N ) is the geodesic diameter of N .Following 2D-GEM [22] , we compare HOPE with the following methods: EM [42] , GE [25] , RF [40] , PFM [39] , FSPM [28] , Kernel-Matching [26] , and GRAMPA [14] , [22] , SGMDS [1] , FM [34] , BIM [24] , Mobius [27] , Best-Conformal [24] , Kernel-Matching [26] , DIR-500 [49] which uses 500 eigenvectors, DIR-1000 [49] which uses 1000 eigenvectors.On ZoomOut, we start with a functional map using the first 20 eigenvectors of the LBO, then we iteratively add an eigenvector until we reach the 120th eigenvector after which we stop.On all other Baselines, we follow the settings from [22] .On HOPE, on all datasets we set the LMD threshold ϵ, staring from ϵ = 100 and 10 equally spaced values to ϵ = 0.2 i.e., we use ϵ = linespace(100, 0.2, 10) and we set t = 60. When the last value of e is reached, it is maintained for the rest of the iterations. We equaly set k max = 8 for all datasets. For the LMD, we used the second ring neighborhood following [49, 22] . Comparison on non-isometric shapes. On the dataset with topological noise TOPKIDS (Figure 5 (a)), the top 3 methods are the noise-robust methods 2D-GEM with 96.7% vertices correctly matched, followed by HOPE with 94.9%, and GRAMPA with accuracy of 84.5%. This shows that 2D-GEM, HOPE, and GRAMPA are indeed adapted for non-isometric shapes. Moreover, as discussed in [22] , GRAMPA is very time-consuming as it needs the full basis of the graph adjacencies of the shapes, while HOPE and 2D-GEM do not. For example on a shape with 12500 vertices from SCAPE, GRAMPA takes roughly 118 seconds on our set up while 2D-GEM and HOPE take 56 and 52 seconds respectively. Better still HOPE is a general model as it can be seen that with the same parameter of ϵ (see Section 5.5) for its LMD, it performs well on all datasets. Comparison on isometric shapes. On isometric shapes, methods that enforce stronger geometric constraints on their refinement pipelines such as 2D-GEM (with appropriate parameters), DIR, and HOPE. HOPE outperform all other baselines. 2D-GEM achieves 92.54% accuracy at geodesic error 0 on the TOSCA dataset, as well as 74.8% accuracy on the SCAPE dataset, while HOPE achieves 80.11% on SCAPE and 92.54% on TOSCA using no matrix decomposition. Third is DIR-1000 [49] with 1000 eigenvectors which achieves around 69.5% accuracy at geodesic error 0 on SCAPE, and 59.8% accuracy at geodesic error 0 on TOSCA. Moreover, unlike DIR (and 2D-GEM,), HOPE generalizes to this isometric setting using the same parameters (see Section 5.5).Comparison on remeshed shapes. On the remeshed datasets FAUST_r, SCAPE_r and TOSCA_r in figure 6 , methods that enforce stronger geometric constraints such as DIR performed poorly, while those that are more robust to noise such as 2D-GEM (with appropriate parameters), ZoomOut and HOPE perform relatively well. Though 2D-GEM does not outperform ZoomOut in this setting (probably due to differences in the mesh connectivity), HOPE still outperforms ZoomOut though by a smaller margin than on datasets with similar mesh connectivity amongst pairs. Time Comparison. Here we compare the time usage per shape for both 2D-GEM, HOPE, and ZoomOut on the TOSCA dataset. It can be seen from figure 7 that HOPE is relatively faster than 2D-GEM, and even ZoomoOut (for the given parameters of ZoomOut that we used for our experiments).Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification: See attached code.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
limitations: 
 The main limitation of this work is the reliance on correlated triangulations between the pair of shapes to be matched as outlined in Section 4.2. One can see this drawback on figures 6 which show that though HOPE still performs well on remeshed shapes (shapes where the triangulations are not very correlated), and even outperforms other baselines, it nonetheless does not perform as well as on figures 5(b), 5(a) and 5(c) where the triangulations of the pairs matched are strongly correlated. This is a major drawback because triangulations are often hard and expensive to get, especially consistent triangulations between shapes.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: See Section 5.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
conclusion: 
 We introduced an effective and easy-to-implement map refinement strategy consisting of; (a) detecting poorly matched vertices (nodes) using the concept of local map distortion (LMD), and (b) improving the map of these poorly matched vertices via noise robust k-hop pairwise descriptors. We then conducted a series of experiments to show that our framework is effective and generalizable to different shape datasets. We also discussed the main limitation of our work (the fact that it is reliant on the triangulation consistency between the shapes matched).Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes] Justification: See Section 5.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
broader impact: 
 This work proposes a map refinement strategy for shape matching that is based on matching different k-hop neighborhoods of vertices. It then validates the effectiveness of this strategy on several shape matching datasets. This can spark new research on the importance of large neighborhood statistics for shape matching and other related tasks. As we focus solely proposing a framework for map refinement for shape matching, we do not see clear negative impact of this work.Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] Justification: See Section 5.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.• The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: This does not apply to our task Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: We implemented the code and cite all packages and dataset used.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 
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 Appendix: A HOPE AlgorithmIn this section, we present the algorithm for HOPE. This is given in algorthm 1Algorithm 1 : HOPE 0. input An initial map T 0 , distance matrices d M and d N , LMD threshold ϵ, maximum iteration t, and the maximum hop k max and k = 1while 0 ≤ i ≤ t do 1. build A M,k and A M,k 2. set k = k + 1 if k > k max then 3. k = 1, end if 4.Given T i , use the LMD to locate well-matched points lmks and poorly-matched points N lmks, 5. Use these N lmks pairs to update the T i (N lmks) by using the GMWM to solve equation ( 7), end while return T .In this section, we conduct parameter and ablation studies. We use the following settings: It can be observed from figure 8 that the best model overall is HOPE-th which is best on isometric (figure 8(b )) shapes and non-isometric shapes (figure 8(a) ) and performaing comparatively to other variants on remeshed shapes (figure 8(c )). This is followed by HOPE. This indicates that the ϵ we used for our main experiments in Section 5 is sub-optimal since we randomly selected the range to be linspace(100, 0.2, 10) without any parameter tuning to demonstrate the effectiveness and generalizability of HOPE. We notice that the variants without the LMD (HOPE-varyhop and HOPE-fixhop) all struggled on the isometric shape (figure 8(b)) validating our observation that the k-hop neighborhood of nodes may not be very unique especially when there are symmetries, and as such using a stronger constraint like the LMD is beneficial in such settings (Section 3 and 4)In this section, we conduct studies on the effects of different initializations on HOPE. We use the following settings:• HOPE-SHOT: where we use SHOT [45] descriptors to initialize t 0 in algorithm 1,• HOPE-HKS: where we use Heat Kernel Signatures (HKS) [8] descriptors to initialize t 0 in algorithm 1, • HOPE-WKS: where we use Wave Kernel Signatures (WKS) [4] descriptors to initialize t 0 in algorithm 1.Figure 9 shows that indeed SHOT [45] which is commonly used in practice is a robust and good descriptor to use as initialization. On the other hand while both the HKS [8] and WKS [4] provided an initialization that could be enhanced by HOPE relatively well on the isometric shape (figure 9(b)), only WKS and SHOT were suitable for the remeshed shape (figure 9 (c)), while only SHOT initializations where relatively okay in terms of accuracy on the non-sometric shape (figure 9 (a)).In this section, we show the performance of HOPE on partial shape matching where we match the full shapes to the partial shapes. Here we used the SHREC16 HOLES and SHREC16 CUTS following Cao et al. [9] . Figure 10 shows that although HOPE was not designed specifically for the partial shape setting, it nonetheless performs relatively well on average as seen by the average geodesic curve.In this section, we show the performance of HOPE on another non-isometric shape dataset SMAL_r Cao et al. [9] , where we perform intra-class matching of 298 pairs of shapes. Figure 11 shows that HOPE again outperforms other baselines even when the mesh triangulations are not strongly correlated (due to noise).In this section, we show the visual comparisons between HOPE and other baselines, on SCAPE_r (figure 12 ), SHREC16 (figure 13 ), and TOPKIDS (figure 14 ). • If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example (a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. (b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. (c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 


Decision: The paper obtained mixed reviews, with two reviewers voting for reject and two voting for accept. On the positive side, good presentation, theoretical contribution (simple and sound methodology) and good performance are noted. On the downside, limited novelty, limited evaluation, and a negative trade-off with smoothness of the resulting correspondence. The extensive rebuttal is acknowledged. In particular, we weighted down one very negative review in light of its divergent complaints regarding the presentation and lack of understandability that could not be corroborated. We urge the authors to still improve the paper regarding the raised concerns, though. Overall, we follow the positive reviews and the paper should be accepted in light of its theoretical contribution and practical advantages.

Reviews: review: paper_summary: paper_summary: The paper focuses on the shape-matching tasks and proposes a shape-matching refinement technique based on the K-hop neighborhood descriptor and local map distortions. The experiments show improved results compared to existing approaches in isometric and non-isometric shape registration.
strength_weakness: strength_weakness: - While there is existing work on deep and modern feature extraction, the experiments in this paper are limited to the SHOT descriptor. Incorporating other classical or learned signatures could demonstrate broader generalizable performance. Additionally, evaluating the impact of the quality of initial matches could provide further insights into the robustness of the proposed method.

- The paper is motivated by the observation that existing approaches often prioritize smoothness over accuracy. Although this paper emphasizes improving accuracy, it raises the question of whether smoothness might be compromised as a result. Detailed analysis and results are needed to assess the balance between these two aspects.
questions: questions: The paper suggests using geodesic error as a loss function to improve smoothness and accuracy. Given that other methods also utilize geodesic error, what aspects of the HOPE approach allow it to leverage geodesic error more effectively and potentially yield superior results? How does this approach manage the balance between local and global geometric consistency?
limitations: limitations: The method might face challenges in handling topological variations between shapes.
paper_summary: paper_summary: This submission proposes a descriptor utilizing k-hop neighborhoods for non-rigid 3D shape matching. The descriptor is used jointly with local map distortion for map refinement. 

Overall, I am highly frustrated during reviewing this submission. While I have tried my best to parse and understand the technical details, the confusing formulations and derivations, together with missing definitions have prevented me from doing so. Overall, I would encourage the authors to revise carefully the submission, so that it is readable.
strength_weakness: strength_weakness: The paper lacks readability, especially on the technical part. An incomplete list of frustrations I have encountered is as follows: 

1. The presentation is poor. Starting from Eqn.(1), I have got lost. There has never been an explicit formula or at least a clear reference to the prior works for W_M(T^t, T^{t-1}). Similarly Q_M, Q_N are not defined. I have tried my best to parse the statements, but have to give up in the end. 

2. Theorem 3.1 and its proof are also confusing. First of all, Q_M and Q_N in proof 3.1 are *not* mentioned in the theorem. Second, functional maps by its definition is defined regarding the basis, instead of some arbitrary vertex-wise descriptor. However, I did not anything around L95 saying Qs are eigenbasis (or any kind of basis regarding C). 

3. In Eqn.(6), the definition of A_M, k(T, T^{k-1}) is again undefined. 

4. The proof of Theorem 4.1 is pointless. Please use the standard mathematical derivation, rather than piling up a series of unverified statements.
questions: questions: 1. I would disagree with the argument in L27. Shape matching methods would rarely aim for solely smoothness without accuracy -- the latter is the most important, while the former can serve as good regularization. 

2. Should niegh(i) in Eqn.(4) be neigh(i)? 

3. In Fig.4, I can not see the curve regarding HOPE. It would be useful to report geodesic error in the legend.

4. If I understood correctly, the appendix should be behind checklist.
limitations: limitations: It looks fine to me.
paper_summary: paper_summary: This paper introduces HOPE, a method that leverages k-hop neighborhoods as pairwise descriptors to achieve both accuracy and smoothness in shape matching. This approach aims to overcome the limitations of existing methods that often struggle to balance these two crucial aspects. The k-hop neighborhood concept involves considering the vertices within a certain distance (k hops) from a given vertex, providing a local context that can be used to compute pairwise descriptors. The authors validate their approach using several benchmark datasets, including SCAPE, TOSCA, and TOPKIDS.
strength_weakness: strength_weakness: 1. Novelty: The methods in the paper lack overall novelty, as the proposed approach involves minor optimizations and modifications based on existing work.
2. Parameter Sensitivity: Although not explicitly discussed in the provided sections, methods that involve iterative refinement and multiple descriptors often require careful tuning of parameters. The performance of HOPE may depend on the selection of parameters such as the number of hops (k) and the weights used in the LMD optimization. A thorough analysis of the method's sensitivity to different parameters would be beneficial, possibly including guidelines or heuristics for parameter selection.
3. Experimental Results: In the compared datasets, this method does not show a significant improvement over the baseline. The advantages of the technique are not evident from the provided qualitative and quantitative results. The paper mentions other works that address smoothness and accuracy but could benefit from a more detailed comparative analysis, showing direct performance metrics against the most recent and relevant methods.
4. Benchmark: There are relatively few results on non-isometric datasets in the comparison. It is suggested to supplement results from other datasets, such as SMAL_r [a], DT4D-H [b], or other data in SHREC07 [c].
[a] Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching.
[b] Smooth Non-Rigid Shape Matching via Effective Dirichlet Energy Optimization.
[c] http://partial.ge.imati.cnr.it.
questions: questions: 1. What is the impact of the initialization map quality on this method, and how does it compare in tolerance level with other baselines?
2. How much does the method improve over the baseline in specific quantitative results on the SHREC16 dataset? There are no qualitative results or specific quantitative results, including geodesic error.
3. There are no quantitative metrics provided to demonstrate the smoothness of the map.
4. What is the value of kmax set in the experiments, and is it consistently optimal across different datasets?
limitations: limitations: Please refer to the Weaknesses and Questions above.
paper_summary: paper_summary: The paper presents a new method for shape correspondence that extracts descriptors that are both smooth over the manifold and distinctive enough to find high precision point matching. One can view the presented method as an improvement over the 2D-GEM method, that, as opposed to the latter, does not use the eigenfunction of the laplacian and interacts with more the 2-hop neighborhood.  It is extensively tested on various benchmarks in the field, near-isometric and non-isometric ones, and presented promising results.
strength_weakness: strength_weakness: The new method to some extent resembles the 2D-GEM method, which limits the novelty. As presented in the limitation, since it uses the vertex neighborhood it is vulnerable to remeshing, and it counters the statements in the paper about the method usefulness in real data scenarios (overclaiming). 
Small issues: I think [1] should be referred to because of its similarly to the ideas presented in the paper, they also use alignment processing of the initial descriptors by reducing geodesic distance error.

[1] Bracha, A., et al., 2020. Shape correspondence by aligning scale-invariant LBO eigenfunctions. 3DOR: Eurographics Workshop on 3D Object Retrieval
questions: questions: see weaknesses
limitations: limitations: see weaknesses

Comments: title: Responce to rebuttal
Responses to "Some final remarks" by Reviewer E6Vo
Some final remarks
Nice paper.
Response 2 to  Reviewer NAeg
Response to Reviewer E6Vo on "Could you please respond with mathematical formulations?"
Could you please respond with mathematical formulations?
Response to Reviewer E6Vo on "On Theorem 4.1"
Response to Reviewer E6Vo on "On the definition of accuracy"
Response to Reviewer E6Vo on "Let us start with formulation"
On Theorem 4.1
On the definition of accuracy
Let us start with formulation
comment: Dear authors,

Thank you for submitting your rebuttal and addressing the concerns raised. I have reviewed the rebuttal and found that my questions have been addressed.

Best regards,
dry5
Dear Reviewer, we are grateful for the time spent to Review our work. We address the concerns and misunderstandings raised in the final remarks below:

**On Theorem 3.1:**

In our response to the Reviewer’s first Reviews we addressed this by pointing out to the Reviewer that we had made a typo here in the proof of the theorem by using Q instead of U as in the definition of the theorem. But this does not change the proof nor the conclusion of the theorem. This theorem proves that all iterative map refinement methods that use truncated basis (such as first k-eigen vectors or singular vectors of some shape descriptors whether pairwise or pointwise) will essentially try to refine the map by aligning clusters of points where with clusters being defined by this truncated basis (since it is well known that the truncated SVD or eigen decomposition are soft clusters). Thus, all these methods will fail at accuracy since it will be matching clusters rather than points. While to some extent in many cases maintaining smoothness depending on which descriptor the truncated basis was obtained from.

**On Theorem 4.1:**

Regarding the proof that the k-hop algorithm will recover the ground truth map under certain conditions, we refer the Reviewer again to Section 4.2, particularly lines 167-175 where we cited [26] and [48] which both give riguouruous and lengthy proofs for using 1 hop or k hop adjacencies for refinement. We then gave Theorem 4.1 that shows using k= 1, 2,3, …, hops for is same as using different hop neighbors as witnesses for the map refinement. As such, we propose to use different hop neighborhoods iteratively which will serve to enforce both local and global consistency in the map (See Section 4.2-4.4).

**On connectivity**

We do not only claim superior performance to ZoomOut but prove this experimentally on most if not all datasets by a hudge margin in some cases. The curves show this as pointed out by the Reviewer. For the Qualitative analysis. We refer the Reviewer to figure 4 in our attached PDF as response to all Reviewers. It clearly shows that ZoomOut fails to be accurate or even be continuous.

**On experimental results**

We refer the Reviewer to the variety of datasets used again and point out that on all these datasets including the ones where the triangulations are not consistent, HOPE (our work) outperforms all baselines in some cases by a significant margin. 
We admitted that HOPE is limited because it relies on the triangulations (in our limitation Section 6). Nonetheless, we showed that even on non-correlated meshes HOPE still outperforms other baselines.

**Additional Remarks**

Given all these, we really believe our work brings a significant contribution to the shape matching community and welcome any further questions from the Reviewer. Moreover, we provided the Source code for HOPE as supplementary material which the Reviewer may use to test other custom meshes (and feel free to give us feedback).
For the sake of fairness, I have re-examined this submission with the clarification made by the authors. Below are my new questions: 

1.	On Theorem 3.1: 

Let alone the confusing purpose of this theorem in the **related work** section, I am confused by the theorem description and the proof. Following the authors’ rebuttal, W in Eqn.(1) denotes pair-wise descriptor such as geodesics, Q in Eqn.(2) denotes vertex-wise descriptor such as SHOT, HKS. 
The claim of theorem 3.1 is “Using **W** for the map refinement via Functional maps helps group nearby clusters together assuming the functional map is perfectly accurate. “. On the other hand, in the proof, **Q** is used throughout. To the best of my knowledge, these two should be assumed to be independent (except for cases where Q is derived from W), right?

2.	On Theorem 4.1: 

In general, a serious theoretical analysis for map refinement method should include at least two perspectives: 1) the proof of convergence; 2) the proof of certain property. For example, Zoomout proves that the (ideal) optimal solution is an isometric map between two shapes. Huang and Guibas [a] prove that their algorithm can recover exact maps under certain noise model (with respect to initial maps). 

Regarding this submission, I am not sure what is the theoretical insight about Theorem 4.1: The claim is about agreement with T^{t-1}, which can be traced back to initialization. As I mentioned above, the expected theoretical analysis should be something like the algorithm can converge to good solution even if initialization is not perfect. To this end, arguing some property aligning with previous iterations seem point-less to me. 

[a] Consistent Shape Maps via Semidefinite Programming, Q. Huang and L. Guibas, SGP, 2013. 

3.	On connectivity

The submission claims superior performance in TOPKIDS. In Fig.2, the error plot is nearly perfect, however the qualitative result seems to be worse than Zoomout. Also, if the method depends on adjacency, then how can it be robust with respect to topological noises, i.e., there are a lot of short cuts in the regarding meshes?

4.	On experimental results

The performance gap between SCAPE and SCAPE_r (Fig.4(b) vs. Fig.5(b)) seems to suggest the proposed method to some extent depends on the unified triangulation prior, which is not promising as the shape matching community has long passed the point caring about matching shapes with uniform triangulation.  

To be honest, I can go on with the question list, but my time could have been contributed to something more meaningful. I have tried my best to understand and evaluate this submission, but can hardly find any evidence for changing my initial rating.
The authors answered our equations, and present a nice methodology to overcome difficulties in shape matching.
*NOVELTY*

Please find below the novelty of HOPE (our work), especially when compared to GEM:

-	First, our proposed HOPE is the first work that uses different K-hops neighborhoods as witnessed for iterative map refinement for shape matching. We justified this in Section 4.2 (supported by theorem 4.1 )
-	Second, we showed in section 4.4 (using theorems 3.1 and 4.1) that GEM does not generalize to different datasets without a significant change in the algorithm. We showed that changing the parameters as they did in their paper is really changing the algorithm, and as such GEM is really two different algorithms from which we choose one depending on the parameters, because choosing large thresholding parameters as they do for non-isometric shapes, re-meshed shapes, and partial shapes deactivates the LMD. This is a huge disadvantage because generally in real life one does not know whether the shapes one is dealing with are isometric or not.

*THEORETICAL PROOF*

We would like the Reviewer to highlight which part of the proof of the usefulness of the k-hop neighborhood was unclear so that we can address this. Since the Reviewer previously did not mention anything about this, else we would have done our best to address this.

*EXPERIMENTAL PERFORMANCE*

With regards to the experiments, the only baseline that looks competitive is indeed GEM, but again:

-	HOPE outperforms GEM in most datasets. GEM does not generalize. The only reason GEM even performs well in our experiments in Section 5 and the Appendix is because we significantly changed their algorithm by using different set of parameters for different datasets as they did in their paper (since choosing large thresholding parameters as they do for non-isometric shapes, re-meshed shapes, and partial shapes deactivates the LMD).  We showed this in Section 4.4 where we used a toy example from the TOPKIDS dataset to prove that if we do not change the parameters, GEM will perform extremely poorly since GEM is really two algorithms from which using different sets of parameters will select one.
-	So really HOPE generalizes to all Datasets while outperforming GEM significantly in the re-meshed (Figure 5) and partial shapes (Figure 9) as well as the SCAPE (Figure 4). Moreover, in addition to outperforming GEM, HOPE also outperforms all other baselines in all experiments (by a large margin in some cases).
- We discussed in Section 4 that the Reason HOPE can generalize are: (a) first that the K-hops are noise robust (thus addressing the re-meshed shapes and noises to some extent), and (b) secondly are more unique than other shape descriptors.

*REPORTING MEAN GEODESIC ERROR*

With regards to reporting the mean geodesic errors:

- In case of acceptance, just like the SMAL_r experiments we added, we can also add these to our Appendix as suggested.
Thanks to the authors for their thorough response and the additional experiments on SMAL_r. I carefully reviewed the authors' replies and the corresponding sections of the paper. However, I believe that the response does not address my main concerns.

First, I think the paper lacks novelty. The authors emphasize using different k-hop neighborhoods, but I believe this is merely an extension of previous work without introducing new insights. Additionally, the theoretical proof provided regarding k-hop is confusing.

Regarding the experiments, specifically Fig. 4, I do not believe the results significantly improved over the baseline. While there seems to be some improvement for remeshed meshes, I suggest discussing the reasons behind these results. Moreover, I do not observe a significant improvement over GEM in most cases in the qualitative examples chosen in both the main text and supplementary materials.

Another suggestion is to provide the mean geodesic error values along with the curves so that the improvement of the method can be directly compared. I believe the entire work requires significant improvements in both writing and innovation, and I do not think this work is ready for publication in NeurIPS 2024.
We apologize to the Reviewer if our answers to the Reviewer’s concerns were not clear.  With hope to change the Reviewer’s scores, please find attached responses in bullets below with as much formulations as possible. We hope they will help with more clarification: 
- W_N \in T^{n * n} is a matrix of pairwise descriptor for vertices n of shape N. It can be geodesic distances or any other pairwise descriptors (In our case we use k-hop neighborhoods as seen in lines 166-202 of our paper)
- T^t and T^{t-1} are the current map and the previous map we want to recover and are \inR^{1*n}. For example, if we have two shapes M and N with 4 vertices each and the map T^{t-1} = [ 2, 1,0, 3], then this means vertices 0,1,2,3 in shape N correspond to vertices 2,1,0,3 (respectively) in shape M according to this map.
- The Trace in the Linear Assignment Problem (LAP): The LAP involves finding a permutation (P) of the columns of a cost matrix ( C ) such that the sum of the corresponding diagonal elements (the trace of the permuted matrix) is minimized or maximized. It is formulated as argmax_P Trace(PC). In our case our cost matrix is C = W_N(:, T^{t-1})*W_M = W_N*P’W_M (where P’ = mat(T^{t-1}) i.e., the matrix form of T^{t-1}). And our LAP is argmax_P Trace(PC) where P = mat(T^{t}) i.e., the matrix form of T^{t}). 
- For accuracy, indeed directly enforcing this constraint in the optimization problem is impossible as the Reviewer rightly said because the ground truth map is not available. As such the accuracy constraint is usually enforced by trying to use accuracy enforcing pairwise or point-wise descriptors as we discussed in Section 3 of the paper.
I have read all responses regarding formulation, accuracy, and Theorem 4.1. And I am still confused, could you please provide answers with self-contained, well-defined derivation to my questions, instead of a pile of sentences? 

For instance, in kernel matching paper, everything is expressed as clear matrix operation, while in your case, you say something like this: 

'W_M(T^t, T^{t−1}) \in R^{n×n} is the pair-wise descriptors of vertices for shape M with its rows and columns aligned using the map T^t and the previous iterations map T^{t−1}'

I have not doubted the correctness of the formulation. The problem is that the above definition is wordy and confusing. How is W_M computed/defined explicitly?

Similarly, I can imagine there is a link between the trace and assignment problem. However, your response again is descriptive. A much better answer would be to explicitly draw connection between such with **equations**. 

On accuracy, the impression I got is that the accuracy you focused on is ultimately not different from the general accuracy (i.e., mean geodesic error), as shown in Fig. 4 and 5. Also an obvious fact is that defining accuracy as percentage of perfect matching is not optimizable in practice -- as the ground-truth annotation is not always available, in fact, most of the time. 

Though I would be happy to discuss and open to be corrected, I would expect efficient, precise communications rather than reading *hand-wavy arguments* and deriving *claimed equivalent forms* myself.
Again, we thank the Reviewer for seeking more clarity from us. In hope to change the Reviewer’s comment we answer as follows: Equation 6 is the well-known Linear assignment problem where we try to find the permutation matrix T^t which maximizes the assignment of rows to columns (trace maximization). This assignment of rows to columns of a cost matrix where the rows are vetices of shape 1 and the columns are vertices of shape 2 is shape matching. And so our theorem 4.1 shows that trying to do shape matching where our cost matrix is the product of corresponding k-hop neighborhoods (given an initial neighborhood aligning map T^t-1) is the same as finding the map T^t that matches nodes with the most nodes in common in their k-hop neighborhood.
We once again thank the Reviewer for this question. We point out that the Reviewer’s scenario has to do with smoothness. Accuracy as from the definition we adopted from the Neurips paper 2D-GEM has to do with matching at geodesic error 0 strictly, while smoothness has to do with matching as we deviate slightly away from geodesic error 0 (i.e., conservation of neighborhoods). We showed that aiming for accuracy while using k-hop neighborhoods (our HOPE) will not only account for accuracy, but also smoothness in our experiments (hence addresses your scenario).   For example, even when shapes do not have consistent triangulations, we showed that HOPE outperforms baselines at matching away from geodesic error 0. This can be seen in the experiments on the re-meshed shape datasets SCAPE_r, FAUST_r, TOSCA_r , and on partial shapes in the paper’s appendix (and the experiment on the non-isometric SMAL_r in our Rebuttal PDF). In all these our proposed HOPE performs comparatively better than other baselines. An explanation on why using k-hop neighborhoods to enforce accuracy also enforces smoothness has to do with the definition of smoothness itself (which in a simplified way can be defined as a preservation of neighborhoods, a fact known also in the field of graph neural networks).
We thank the Reviewer for being willing to ask us for more clarity. In hope of changing the Reviewer’s scores we respond as follows. For equation (1) of our paper we refer the Reviewer to equations 13, 14, 15 and 16 of the CVPR paper “Efficient Deformable Shape Correspondence via Kernel Matching”. Or the ICLR paper. " DEEP GRAPH MATCHING CONSENSUS" in equation 6. Moreover, equation (1) can still be obtained from papers such as the Neurips paper   2D-GEM (“Exact Shape Correspondence via 2D Graph Convolution”), by explicitly writing out their Cp in algorithm 1 using their definition of Cp in equation 10 and Zl in equation 9. This formulation is true for all iterative refinement algorithms that use pairwise descriptors.
I have to admit that I have no clue how the proof is going on, as I have trouble to understand the formulation from the beginning. 

The obvious observation is that the objective of Eqn.(6) is entirely absent from your proof. Where is the Trace going?
A quick question: how does your definition of accuracy (i.e., %Correspondences as geodesic error 0) make sense at all? For instance, I have a map with 50% points at error 0 + 50% points at error 0.1, and a map with 50% points at error 0 + 50% points at error 1.0, the two are equally good under your definition, is that sensible? 

Beyond that, how can you set the percentage of perfect matching as an optimization goal? It is discrete...
Still, I do not see a self-contained formulation/explanation regarding Eqn.(1). Could you please kindly refer to a specific formulation in **any** of the referred papers? 

I got the idea that it is an iterative refinement, it is the undefined notations in Eqn.(1) that makes me confusing and frustrating...