(also known as Hyperdimensional Computing). This framework is well suited for implementation in stochastic, emerging hardware and it naturally expresses the types of cognitive operations required for Artificial Intelligence (AI). We demonstrate in this article that the field-like algebraic structure of Vector Symbolic Architectures offers simple but powerful operations on high-dimensional vectors that can support all data structures and manipulations relevant to modern computing. In addition, we illustrate the distinguishing feature of Vector Symbolic Architectures, "computing in superposition," which sets it apart from conventional computing. It also opens the door to efficient solutions to the difficult combinatorial search problems inherent in AI applications. We sketch ways of demonstrating that Vector Symbolic Architectures are computationally universal. We see them acting as a framework for computing with distributed representations that can play a role of an abstraction layer for emerging computing hardware. This article serves as a reference for computer architects by illustrating the philosophy behind Vector Symbolic Architectures, techniques of distributed computing with them, and their relevance to emerging computing hardware, such as neuromorphic computing.

The ability to encode and manipulate data structures with distributed neural representations could qualitatively enhance the capabilities of traditional neural networks by supporting rule-based symbolic reasoning, a central property of cognition. Here we show how this may be accomplished within the framework of Vector Symbolic Architectures (VSAs) (Plate, 1991; Gayler, 1998; Kanerva, 1996), whereby data structures are encoded by combining high-dimensional vectors with operations that together form an algebra on the space of distributed representations. In particular, we propose an efficient solution to a hard combinatorial search problem that arises when decoding elements of a VSA data structure: the factorization of products of multiple codevectors. Our proposed algorithm, called a resonator network, is a new type of recurrent neural network that interleaves VSA multiplication operations and pattern completion. We show in two examples-parsing of a tree-like data structure and parsing of a visual scene-how the factorization problem arises and how the resonator network can solve it. More broadly, resonator networks open the possibility of applying VSAs to myriad artificial intelligence problems in real-world domains. The companion article in this issue (Kent, Frady, Sommer, & Olshausen, 2020) presents a rigorous analysis and evaluation of the performance of resonator networks, showing it outperforms alternative approaches.

We develop theoretical foundations of resonator networks, a new type of recurrent neural network introduced in Frady, Kent, Olshausen, and Sommer (2020), a companion article in this issue, to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a resonator network can efficiently decompose the composite into these factors. We compare the performance of resonator networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that resonator networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and searching in superposition, by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of resonator networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing the guarantee of global convergence, resonator networks are dramatically more effective at finding factorizations than all alternative approaches considered.