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.

本文研究了二进制高维计算（HDC）的几何问题，发现了由 HDC 绑定运算符引起的相似性结构，并展示了拉普拉斯核在该设置中的应用，激励我们提出了新的编码方法 Laplace-HDC，改进了先前的方法。我们描述了这些结果如何表明二进制 HDC 在对图像的空间信息编码方面存在局限性，并讨论了潜在的解决方案，包括使用 Haar 卷积特征和定义平移等变 HDC 编码。文中还呈现了几组数值实验，突出了 Laplace-HDC 与其他方法相比的改进精确性。我们还对所提出的框架的其他方面进行了数值研究，如鲁棒性和基本的平移等变编码。