We develop theoretical foundations of Resonator Networks, a new type of recurrent neural network introduced in Frady et al. (2020) 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 this guarantee of global convergence, Resonator Networks are dramatically more effective at finding factorizations than all alternative approaches considered.
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