Despite recent advancements in deep neural networks for point cloud recognition, real-world safety-critical applications present challenges due to unavoidable data corruption. Current models often fall short in generalizing to unforeseen distribution shifts. In this study, we harness the inherent set property of point cloud data to introduce a novel critical subset identification (CSI) method, aiming to bolster recognition robustness in the face of data corruption. Our CSI framework integrates two pivotal components: density-aware sampling (DAS) and self-entropy minimization (SEM), which cater to static and dynamic CSI, respectively. DAS ensures efficient robust anchor point sampling by factoring in local density, while SEM is employed during training to accentuate the most salient point-to-point attention. Evaluations reveal that our CSI approach yields error rates of 18.4\% and 16.3\% on ModelNet40-C and PointCloud-C, respectively, marking a notable improvement over state-of-the-art methods by margins of 5.2\% and 4.2\% on the respective benchmarks. Code is available at \href{https://github.com/masterwu2115/CSI/tree/main}{https://github.com/masterwu2115/CSI/tree/main}

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