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Vector space models for symbolic processing that encode symbols by randomvectors have been proposed in cognitive science and connectionist communitiesunder the names Vector Symbolic Architecture (VSA), and, synonymously,Hyperdimensional (HD) computing. In this paper, we generalize VSAs to functionspaces by mapping continuous-valued data into a vector space such that theinner product between the representations of any two data points represents asimilarity kernel. By analogy to VSA, we call this new function encoding andcomputing framework Vector Function Architecture (VFA). In VFAs, vectors canrepresent individual data points as well as elements of a function space (areproducing kernel Hilbert space). The algebraic vector operations, inheritedfrom VSA, correspond to well-defined operations in function space. Furthermore,we study a previously proposed method for encoding continuous data, fractionalpower encoding (FPE), which uses exponentiation of a random base vector toproduce randomized representations of data points and fulfills the kernelproperties for inducing a VFA. We show that the distribution from whichelements of the base vector are sampled determines the shape of the FPE kernel,which in turn induces a VFA for computing with band-limited functions. Inparticular, VFAs provide an algebraic framework for implementing large-scalekernel machines with random features, extending Rahimi and Recht, 2007.Finally, we demonstrate several applications of VFA models to problems in imagerecognition, density estimation and nonlinear regression. Our analyses andresults suggest that VFAs constitute a powerful new framework for representingand manipulating functions in distributed neural systems, with myriadapplications in artificial intelligence.

Computing on Functions Using Randomized Vector Representations