# An Application of Manifold Learning in Global Shape Descriptors
## 5 Discussion
The computation cost increases quadratically with \(n\) as does the size of matrices. Running the experiments on a desktop with system specifications mentioned on Section 4, the average total time is 1.18 and 2.29 s with Shape-DNA and LESI algorithms as well as 0.41 and 0.90 s with Shape-DNA and LESI algorithms using TOSCA and McGill datasets, respectively. However, the Shape-DNA code for computing the first part is accelerated by using C MEX files, which makes it incomparable with our method. Therefore, we timed solving the generalized eigenvalue problem, reported as follows: on average it takes 1 and 0.97 s with Shape-DNA and LESI algorithms using TOSCA dataset (\(p\)-value \(=0.64\), not statistically significant), as well as 0.35 and 0.40 s with Shape-DNA and LESI algorithms using McGill dataset (\(p\)-value \(<0.05\), statistically significant). We compared our method with Shape-DNA as it is the basis of cotangent-based descriptors. Other cotangent-based methods have one or more extra steps, which add fraction of seconds. The time difference between these methods are negligible when descriptors are computed using a system with enough RAM, which is easily accessible in today's world. Therefore, with respect to the computational cost, neither of these methods has advantage over the other. In the original Laplacian Eigenmaps, the high dimension data require a considerable amount of processing as the list of all connections need to be computed for the dataset. In fact, for each point in the high dimension space, a given number of nearest neighbors need to be extracted which could be challenging and unmanageable. However, applying this technique to the 3D meshes, we skip this step as the neighbors are already defined and given in the mesh structure. This work benefits from the Laplacian Eigenmap technique in a space in which the vicinities are given. LESI takes advantage of simple Laplacian computation, to form the LB operator, which provides concise and informative shape descriptors. Experimental results prove that LESI is more effective compared with the other powerful descriptors. One limitation of LESI is related to the original Laplacian Eigenmap algorithm introduced by [PERSON] and [PERSON], in which the generalized eigenvalue problem was solved without specifying a boundary condition. It is not clear how the algorithm can handle manifolds with boundaries. Therefore, we limited our experiments to 3D shapes with closed manifolds. Not necessarily a limitation, but a matter of concern, is how the algorithm works in differentiating different models of men ([PERSON] and [PERSON]). While parameter \(t\) provides one degree of freedom for the algorithm, it is data-dependent and needs to be assigned carefully using cross-validation or the validation set approach. It accepts a wide range of values and no unique value is required. Although we investigated only the application of Laplacian Eigenmap in introducing a shape descriptor, there are some other spectral-based manifold learning methods, such as Isomap, LLE, and Diffusion map, which have not been examined. Another possible future work can be investigating the possibility of extending the current contribution to other data modalities, e.g., classification of 2D images/sketches. One can construct a graph from a single 2D image by extracting features from all or sampled pixels. Such an approach benefits from the fact that there is no limit to the number of extracted features when using Laplacian Eigenmap. In the end, even though the focus of this research has been on spectral shape descriptors, we would like to briefly discuss where our method can stand in comparison with learning-based methods. Recently, we have witnessed that deep learning methods have superseded traditional methods in many applications. Several deep learning techniques, (e.g., [59; 60; 61]) have been introduced in this area. The process of training a neural network is very time consuming and requires high performance computing units, i.e., a GPU.