Point clouds represent 3D object geometry as sets of spatial coordinates. A fundamental task in their analysis is keypoint detection, i.e., identifying a sparse subset of points that captures the most salient features. While supervised methods rely on manual annotations, unsupervised approaches leverage intrinsic geometric properties to localize keypoints without human input. We introduce GEKey, a self-supervised framework that uses the geodesic eccentricity transform, an SE(3)-invariant geometric measure, to identify instance-specific salient points. These points serve as pseudo-labels to train a model that aligns its predictions with the object's intrinsic structure. GEKey is simple, efficient, and robust, preserving intra-class variation by balancing instance-specific and category-level structure. Extensive experiments on ShapeNetCoreV2, KeypointNet, CAPE, ITOP, and DeformingThings4D show state-of-the-art performance in both keypoint detection and alignment, with strong resilience to noise, downsampling, deformations, and real-world conditions. The code and pretrained models are available at https://github.com/Pierreoo/GEKey.

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