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Deep learning has shown significant value in automating radiological diagnostics but can be limited by a lack of generalizability to external datasets. Leveraging the geometric principles of non-Euclidean space, certain geometric deep learning approaches may offer an alternative means of improving model generalizability. This study investigates the potential advantages of hyperbolic convolutional neural networks (HCNNs) over traditional convolutional neural networks (CNNs) in neuroimaging tasks. We conducted a comparative analysis of HCNNs and CNNs across various medical imaging modalities and diseases, with a focus on a compiled multi-modality neuroimaging dataset. The models were assessed for their performance parity, robustness to adversarial attacks, semantic organization of embedding spaces, and generalizability. Zero-shot evaluations were also performed with ischemic stroke non-contrast CT images. HCNNs matched CNNs' performance in less complex settings and demonstrated superior semantic organization and robustness to adversarial attacks. While HCNNs equaled CNNs in out-of-sample datasets identifying Alzheimer's disease, in zero-shot evaluations, HCNNs outperformed CNNs and radiologists. HCNNs deliver enhanced robustness and organization in neuroimaging data. This likely underlies why, while HCNNs perform similarly to CNNs with respect to in-sample tasks, they confer improved generalizability. Nevertheless, HCNNs encounter efficiency and performance challenges with larger, complex datasets. These limitations underline the need for further optimization of HCNN architectures. HCNNs present promising improvements in generalizability and resilience for medical imaging applications, particularly in neuroimaging. Despite facing challenges with larger datasets, HCNNs enhance performance under adversarial conditions and offer better semantic organization, suggesting valuable potential in generalizable deep learning models in medical imaging and neuroimaging diagnostics. [ABSTRACT FROM AUTHOR]