Capacity Analysis of Vector Symbolic Architectures

TL;DR

This paper provides a theoretical analysis of the representation capacities of various vector symbolic architectures (VSAs), establishing bounds and connections to other computational models like sketching and Bloom filters.

Contribution

It introduces new bounds on VSA capacities and explores a novel Hopfield network variant for symbolic tasks, linking VSAs to sketching and Bloom filters.

Findings

Derived bounds on VSA vector dimensions for symbolic tasks

Analyzed a Hopfield network variant for similar symbolic tasks

Established connections between VSAs, sketching, and Bloom filters

Abstract

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.