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With the growth of 3D sensing technology, the deep learning system for 3D point clouds has become increasingly important, especially in applications such as autonomous vehicles where safety is a primary concern. However, there are growing concerns about the reliability of these systems when they encounter noisy point clouds, either occurring naturally or introduced with malicious intent. This paper highlights the challenges of point cloud classification posed by various forms of noise, from simple background noise to malicious adversarial/backdoor attacks that can intentionally skew model predictions. While there's an urgent need for optimized point cloud denoising, current point outlier removal approaches, an essential step for denoising, rely heavily on handcrafted strategies and are not adapted for higher-level tasks, such as classification. To address this issue, we introduce an innovative point outlier cleansing method that harnesses the power of downstream classification models. Using gradient-based attribution analysis, we define a novel concept: point risk. Drawing inspiration from tail risk minimization in finance, we recast the outlier removal process as an optimization problem, named PointCVaR. Extensive experiments show that our proposed technique not only robustly filters diverse point cloud outliers but also consistently and significantly enhances existing robust methods for point cloud classification. A notable feature of our approach is its effectiveness in defending against the latest threat of backdoor attacks in point clouds.