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In recent years, Convolutional Neural Networks (CNN) have proven to be efficient analysis tools for processing point clouds, e.g., for reconstruction, segmentation and classification. In this paper, we focus on the classification of edges in point clouds, where both edges and their surrounding are described. We propose a new parameterization adding to each point a set of differential information on its surrounding shape reconstructed at different scales. These parameters, stored in a Scale-Space Matrix (SSM), provide a well suited information from which an adequate neural network can learn the description of edges and use it to efficiently detect them in acquired point clouds. After successfully applying a multi-scale CNN on SSMs for the efficient classification of edges and their neighborhood, we propose a new lightweight neural network architecture outperforming the CNN in learning time, processing time and classification capabilities. Our architecture is compact, requires small learning sets, is very fast to train and classifies millions of points in seconds.