Optimal quadratic binding for relational reasoning in vector symbolic neural architectures

TL;DR

This paper investigates optimal quadratic binding matrices for relational reasoning in neural architectures, introducing octonion-based matrices that improve unbinding accuracy with few pairs and analyzing their performance relative to other methods.

Contribution

It introduces a novel class of quadratic binding matrices based on octonion algebra, demonstrating their effectiveness in relational decoding tasks.

Findings

Octonion-based matrices enable more accurate unbinding with few pairs.

Random quadratic binding performs similarly with many pairs.

Numerical optimization converges to octonion binding.

Abstract

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous works introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs as well as the octonion and previously-proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain.