V ector Symbolic Architectures (VSAs) are a unique approach to performing symbolic style AI.

Vector Symbolic Architectures (VSAs) are one approach to developing Neuro-symbolic AI, where two vectors in \mathbb{R} d are'bound' together to produce a new vector in the same space. VSAs support the commutativity and associativity of this binding operation, along with an inverse operation, allowing one to construct symbolic-style manipulations over real-valued vectors. Most VSAs were developed before deep learning and automatic differentiation became popular and instead focused on efficacy in hand-designed systems. In this work, we introduce the Hadamard-derived linear Binding (HLB), which is designed to have favorable computational efficiency, and efficacy in classic VSA tasks, and perform well in differentiable systems.

At the risk of overstating the case, connectionist approaches to machine learning, i.e. neural networks, are enjoying a small vogue right now. However, these methods require large volumes of data and produce models that are uninterpretable to humans. An alternative framework that is compatible with neural networks and gradient-based learning, but explicitly models compositionality, is Vector Symbolic Architectures (VSAs). VSAs are a family of algebras on high-dimensional vector representations. They arose in cognitive science from the need to unify neural processing and the kind of symbolic reasoning that humans perform. While machine learning methods have benefited from category theoretical analyses, VSAs have not yet received similar treatment. In this paper, we present a first attempt at applying category theory to VSAs. Specifically, we conduct a brief literature survey demonstrating the lacking intersection of these two topics, provide a list of desiderata for VSAs, and propose that VSAs may be understood as a (division) rig in a category enriched over a monoid in Met (the category of Lawvere metric spaces). This final contribution suggests that VSAs may be generalised beyond current implementations. It is our hope that grounding VSAs in category theory will lead to more rigorous connections with other research, both within and beyond, learning and cognition.

Vector Symbolic Architectures (VSAs) are one approach to developing Neuro-symbolic AI, where two vectors in $\mathbb{R}^d$ are `bound' together to produce a new vector in the same space. VSAs support the commutativity and associativity of this binding operation, along with an inverse operation, allowing one to construct symbolic-style manipulations over real-valued vectors. Most VSAs were developed before deep learning and automatic differentiation became popular and instead focused on efficacy in hand-designed systems. In this work, we introduce the Hadamard-derived linear Binding (HLB), which is designed to have favorable computational efficiency, and efficacy in classic VSA tasks, and perform well in differentiable systems. Code is available at https://github.com/FutureComputing4AI/Hadamard-derived-Linear-Binding

While Vector Symbolic Architectures (VSAs) are promising for modelling spatial cognition, their application is currently limited to artificially generated images and simple spatial queries. We propose VSA4VQA - a novel 4D implementation of VSAs that implements a mental representation of natural images for the challenging task of Visual Question Answering (VQA). VSA4VQA is the first model to scale a VSA to complex spatial queries. Our method is based on the Semantic Pointer Architecture (SPA) to encode objects in a hyperdimensional vector space. To encode natural images, we extend the SPA to include dimensions for object's width and height in addition to their spatial location. To perform spatial queries we further introduce learned spatial query masks and integrate a pre-trained vision-language model for answering attribute-related questions. We evaluate our method on the GQA benchmark dataset and show that it can effectively encode natural images, achieving competitive performance to state-of-the-art deep learning methods for zero-shot VQA.

This paper develops an innovative method that enables neural networks to generate and utilize knowledge graphs, which describe their concept-level knowledge and optimize network parameters through alignment with human-provided knowledge. This research addresses a gap where traditionally, network-generated knowledge has been limited to applications in downstream symbolic analysis or enhancing network transparency. By integrating a novel autoencoder design with the Vector Symbolic Architecture (VSA), we have introduced auxiliary tasks that support end-to-end training. Our approach eschews traditional dependencies on ontologies or word embedding models, mining concepts from neural networks and directly aligning them with human knowledge. Experiments show that our method consistently captures network-generated concepts that align closely with human knowledge and can even uncover new, useful concepts not previously identified by humans. This plug-and-play strategy not only enhances the interpretability of neural networks but also facilitates the integration of symbolic logical reasoning within these systems.

Vector Symbolic Architectures (VSAs) have emerged as a novel framework for enabling interpretable machine learning algorithms equipped with the ability to reason and explain their decision processes. The basic idea is to represent discrete information through high dimensional random vectors. Complex data structures can be built up with operations over vectors such as the "binding" operation involving element-wise vector multiplication, which associates data together. The reverse task of decomposing the associated elements is a combinatorially hard task, with an exponentially large search space. The main algorithm for performing this search is the resonator network, inspired by Hopfield network-based memory search operations. In this work, we introduce a new variant of the resonator network, based on self-attention based update rules in the iterative search problem. This update rule, based on the Hopfield network with log-sum-exp energy function and norm-bounded states, is shown to substantially improve the performance and rate of convergence. As a result, our algorithm enables a larger capacity for associative memory, enabling applications in many tasks like perception based pattern recognition, scene decomposition, and object reasoning. We substantiate our algorithm with a thorough evaluation and comparisons to baselines.

Vector Symbolic Architectures (VSAs) give a way to represent a complex object as a single fixed-length vector, so that similar objects have similar vector representations. These vector representations then become easy to use for machine learning or nearest-neighbor search. We review a previously proposed VSA method, MBAT (Matrix Binding of Additive Terms), which uses multiplication by random matrices for binding related terms. However, multiplying by such matrices introduces instabilities which can harm performance. Making the random matrices be orthogonal matrices provably fixes this problem. With respect to larger scale applications, we see how to apply MBAT vector representations for any data expressed in JSON. JSON is used in numerous programming languages to express complex data, but its native format appears highly unsuited for machine learning. Expressing JSON as a fixed-length vector makes it readily usable for machine learning and nearest-neighbor search. Creating such JSON vectors also shows that a VSA needs to employ binding operations that are non-commutative. VSAs are now ready to try with full-scale practical applications, including healthcare, pharmaceuticals, and genomics. Keywords: MBAT (Matrix Binding of Additive Terms), VSA (Vector Symbolic Architecture), HDC (Hyperdimensional Computing), Distributed Representations, Binding, Orthogonal Matrices, Recurrent Connections, Machine Learning, Search, JSON, VSA Applications