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Background: Degenerative spinal pathologies are highly prevalent among the elderly population. Timely diagnosis of osteoporotic fractures and other degenerative deformities enables proactive measures to mitigate the risk of severe back pain and disability. Methods: We explore the use of shape auto-encoders for vertebrae, advancing the state of the art through robust automatic segmentation models trained without fracture labels and recent geometric deep learning techniques. Our shape auto-encoders are pre-trained on a large set of vertebrae surface patches. This pre-training step addresses the label scarcity problem faced when learning the shape information of vertebrae for fracture detection from image intensities directly. We further propose a novel shape decoder architecture: the point-based shape decoder. Results: Employing segmentation masks that were generated using the TotalSegmentator, our proposed method achieves an AUC of 0.901 on the VerSe19 testset. This outperforms image-based and surface-based end-to-end trained models. Our results demonstrate that pre-training the models in an unsupervised manner enhances geometric methods like PointNet and DGCNN. Conclusion: Our findings emphasize the advantages of explicitly learning shape features for diagnosing osteoporotic vertebrae fractures. This approach improves the reliability of classification results and reduces the need for annotated labels. [ABSTRACT FROM AUTHOR]