Representing Sets as Summed Semantic Vectors

TL;DR

This paper demonstrates how a specialized technique can exactly recover individual vectors and their weights from a summed semantic vector, enabling set representation and reasoning in high-dimensional spaces.

Contribution

It introduces a method for exact recovery of set components from summed vectors, advancing vector-based semantic reasoning.

Findings

Exact recovery of set vectors is possible under certain conditions.

The method characterizes the maximum number of vectors recoverable.

Applications include improved semantic reasoning with summed vectors.

Abstract

Representing meaning in the form of high dimensional vectors is a common and powerful tool in biologically inspired architectures. While the meaning of a set of concepts can be summarized by taking a (possibly weighted) sum of their associated vectors, this has generally been treated as a one-way operation. In this paper we show how a technique built to aid sparse vector decomposition allows in many cases the exact recovery of the inputs and weights to such a sum, allowing a single vector to represent an entire set of vectors from a dictionary. We characterize the number of vectors that can be recovered under various conditions, and explore several ways such a tool can be used for vector-based reasoning.