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With the rapid expansion of applied 3D computational vision, shape descriptors have become increasingly important for a wide variety of applications and objects from molecules to planets. Appropriate shape descriptors are critical for accurate (and efficient) shape retrieval and 3D model classification. Several spectral-based shape descriptors have been introduced by solving various physical equations over a 3D surface model. In this paper, for the first time, we incorporate a specific <i>manifold learning</i> technique, introduced in statistics and machine learning, to develop a global, spectral-based shape descriptor in the computer graphics domain. The proposed descriptor utilizes the <i>Laplacian Eigenmap</i> technique in which the Laplacian eigenvalue problem is discretized using an exponential weighting scheme. As a result, our descriptor eliminates the limitations tied to the existing spectral descriptors, namely dependency on triangular mesh representation and high intra-class quality of 3D models. We also present a straightforward normalization method to obtain a scale-invariant and noise-resistant descriptor. The extensive experiments performed in this study using two standard 3D shape benchmarks&#8212;high-resolution TOSCA and McGill datasets&#8212;demonstrate that the present contribution provides a highly discriminative and robust shape descriptor under the presence of a high level of noise, random scale variations, and low sampling rate, in addition to the known isometric-invariance property of the Laplace&#8722;Beltrami operator. The proposed method significantly outperforms state-of-the-art spectral descriptors in shape retrieval and classification. The proposed descriptor is limited to closed manifolds due to its inherited inability to accurately handle manifolds with boundaries.