This paper studies the geometry of binary hyperdimensional computing (HDC), a computational scheme in which data are encoded using high-dimensional binary vectors. We establish a result about the similarity structure induced by the HDC binding operator and show that the Laplace kernel naturally arises in this setting, motivating our new encoding method Laplace-HDC, which improves upon previous methods. We describe how our results indicate limitations of binary HDC in encoding spatial information from images and discuss potential solutions, including using Haar convolutional features and the definition of a translation-equivariant HDC encoding. Several numerical experiments highlighting the improved accuracy of Laplace-HDC in contrast to alternative methods are presented. We also numerically study other aspects of the proposed framework such as robustness and the underlying translation-equivariant encoding.

翻译：本文研究二进制超维计算（HDC）的几何结构，该计算方案使用高维二进制向量对数据进行编码。我们证明了HDC绑定算子所诱导的相似性结构的一个结果，并表明在该场景下自然出现拉普拉斯核，从而激发了我们的新编码方法Laplace-HDC，该方法在先前方法基础上有所改进。我们描述了结果如何表明二进制HDC在编码图像空间信息方面的局限性，并讨论了潜在解决方案，包括使用Haar卷积特征和定义平移等变HDC编码。通过若干数值实验，我们展示了Laplace-HDC相较于替代方法的精度提升。此外，我们还在数值上研究了所提出框架的其他方面，如鲁棒性和底层平移等变编码。