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（Hyper-Dimensional Function Encoding, HDFE）。给定连续对象（如函数）的样本，HDFE能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HDFE能够一致地编码连续对象而无需考虑其采样方式，从而使神经网络能够将连续对象作为机器学习的输入（如分类和回归任务）。此外，HDFE无需任何训练，且已被证明可将对象映射至结构化的嵌入空间，从而促进下游任务的训练。同时，该编码具有可解码性，使神经网络能够通过对编码进行回归来实现对连续对象的回归。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，原始HDFE在性能上已达到与当前最先进算法相当的水平。我们还将HDFE应用于点云表面法向量估计，通过将PointNet简单替换为HDFE，在两个基准测试中分别实现了12%和15%的错误率降低。此外，将HDFE集成至基于PointNet的最新网络（SOTA）后，我们在相同基准测试中将SOTA基线分别提升了2.5%和1.7%。