Capacity Analysis of Vector Symbolic Architectures

Hyperdimensional computing (HDC) is a biologically-inspired framework that uses high-dimensional vectors and various vector operations to represent and manipulate symbols. The ensemble of a particular vector space and two vector operations (one addition-like for "bundling" and one outer-product-like for "binding") form what is called a "vector symbolic architecture" (VSA). While VSAs have been employed in numerous applications and studied empirically, many theoretical questions about VSAs remain open. We provide theoretical analyses for the *representation capacities* of three popular VSAs: MAP-I, MAP-B, and Binary Sparse. Representation capacity here refers to upper 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 $|S \cap T|$) to a given degree of accuracy. We also describe a relationship between the MAP-I VSA to Hopfield networks, which are simple models of associative memory, and analyze the ability of Hopfield networks to perform some of the same tasks that are typically asked of VSAs. Our analysis of MAP-I casts the VSA vectors as the outputs of *sketching* (dimensionality reduction) algorithms such as the Johnson-Lindenstrauss transform; this provides a clean, simple framework for obtaining bounds on MAP-I's representation capacity. We also provide, to our knowledge, the first analysis of testing set membership in a bundle of general pairwise bindings from MAP-I. Binary sparse VSAs are well-known to be related to Bloom filters; we give analyses of set intersection for Bloom and Counting Bloom filters. Our analysis of MAP-B and Binary Sparse bundling include new applications of several concentration inequalities.