We propose Hyper-Dimensional Function Encoding (HDFE). Given samples of a continuous object (e.g. a function), HDFE produces an explicit vector representation of the given object, invariant to the sample distribution and density. Sample distribution and density invariance enables HDFE to consistently encode continuous objects regardless of their sampling, and therefore allows neural networks to receive continuous objects as inputs for machine learning tasks, such as classification and regression. Besides, HDFE does not require any training and is proved to map the object into an organized embedding space, which facilitates the training of the downstream tasks. In addition, the encoding is decodable, which enables neural networks to regress continuous objects by regressing their encodings. Therefore, HDFE serves as an interface for processing continuous objects. We apply HDFE to function-to-function mapping, where vanilla HDFE achieves competitive performance as the state-of-the-art algorithm. We apply HDFE to point cloud surface normal estimation, where a simple replacement from PointNet to HDFE leads to immediate 12% and 15% error reductions in two benchmarks. In addition, by integrating HDFE into the PointNet-based SOTA network, we improve the SOTA baseline by 2.5% and 1.7% in the same benchmarks.

翻译：我们提出超维函数编码（HDFE）。给定连续对象（如函数）的样本，HDFE能够生成该对象的显式向量表示，且该表示对样本分布和密度具有不变性。样本分布与密度不变性使得HDFE能够一致地编码连续对象而无论其采样方式如何，从而允许神经网络将连续对象作为输入执行分类、回归等机器学习任务。此外，HDFE无需任何训练，且被证明能将对象映射到结构化的嵌入空间中，从而促进下游任务的训练。同时，该编码具有可解码性，使得神经网络能够通过回归其编码来回归连续对象。因此，HDFE可作为处理连续对象的接口。我们将HDFE应用于函数到函数的映射任务，其中基础HDFE取得了与最先进算法相匹敌的性能。我们将HDFE应用于点云表面法线估计任务，在两项基准测试中，仅将PointNet简单替换为HDFE便分别带来了12%和15%的误差降低。此外，通过将HDFE集成到基于PointNet的最先进网络中，我们在相同基准测试中分别提升了2.5%和1.7%的基线性能。