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We present a novel attention-based mechanism to learn enhanced point features for point cloud processing tasks, e.g., classification and segmentation. Unlike prior works, which were trained to optimize the weights of a pre-selected set of attention points, our approach learns to locate the best attention points to maximize the performance of a specific task, e.g., point cloud classification. Importantly, we advocate the use of single attention point to facilitate semantic understanding in point feature learning. Specifically, we formulate a new and simple convolution, which combines convolutional features from an input point and its corresponding learned attention point, or LAP, for short. Our attention mechanism can be easily incorporated into state-of-the-art point cloud classification and segmentation networks. Extensive experiments on common benchmarks such as ModelNet40, ShapeNetPart, and S3DIS all demonstrate that our LAP-enabled networks consistently outperform the respective original networks, as well as other competitive alternatives, which employ multiple attention points, either pre-selected or learned under our LAP framework.