Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership $i \in S$ and estimating set intersection sizes $|X \cap Y|$ for two sets of symbols $X$ and $Y$, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

翻译：超维计算（HDC）是一种受生物学启发的框架，它用高维向量表示符号，并通过向量运算对其进行操作。特定向量空间与一组预设向量运算（包括一种类似加法的“捆绑”运算和一种类似外积的“绑定”运算）共同构成*向量符号架构*（VSA）。尽管VSA已在众多应用中得到采用并经过实证研究，但关于VSA的许多理论问题仍有待探索。本文分析了四种常见VSA的*表示容量*：MAP-I、MAP-B，以及两种基于稀疏二值向量的VSA。此处的“表示容量”指执行特定符号任务时VSA向量所需维度的界限，例如在给定精度下测试集合成员关系 $i \in S$ 或估计两个符号集合 $X$ 和 $Y$ 的交集大小 $|X \cap Y|$。我们还分析了一种新型Hopfield网络（一种简单的联想记忆模型）执行部分通常由VSA承担的任务的能力。除了提供VSA容量的新界限外，我们的分析还建立并利用了VSA、 “素描”（降维）算法与布隆过滤器之间的联系。