Christopher J. Kymn

Christopher J. Kymn, Denis Kleyko, E. Paxon Frady et al.

We introduce Residue Hyperdimensional Computing, a computing framework that unifies residue number systems with an algebra defined over random, high-dimensional vectors. We show how residue numbers can be represented as high-dimensional vectors in a manner that allows algebraic operations to be performed with component-wise, parallelizable operations on the vector elements. The resulting framework, when combined with an efficient method for factorizing high-dimensional vectors, can represent and operate on numerical values over a large dynamic range using vastly fewer resources than previous methods, and it exhibits impressive robustness to noise. We demonstrate the potential for this framework to solve computationally difficult problems in visual perception and combinatorial optimization, showing improvement over baseline methods. More broadly, the framework provides a possible account for the computational operations of grid cells in the brain, and it suggests new machine learning architectures for representing and manipulating numerical data.

Christopher J. Kymn, Sonia Mazelet, Annabel Ng et al.

We propose a system for visual scene analysis and recognition based on encoding the sparse, latent feature-representation of an image into a high-dimensional vector that is subsequently factorized to parse scene content. The sparse feature representation is learned from image statistics via convolutional sparse coding, while scene parsing is performed by a resonator network. The integration of sparse coding with the resonator network increases the capacity of distributed representations and reduces collisions in the combinatorial search space during factorization. We find that for this problem the resonator network is capable of fast and accurate vector factorization, and we develop a confidence-based metric that assists in tracking the convergence of the resonator network.

E. Paxon Frady, Denis Kleyko, Christopher J. Kymn et al.

Vector space models for symbolic processing that encode symbols by random vectors have been proposed in cognitive science and connectionist communities under the names Vector Symbolic Architecture (VSA), and, synonymously, Hyperdimensional (HD) computing. In this paper, we generalize VSAs to function spaces by mapping continuous-valued data into a vector space such that the inner product between the representations of any two data points represents a similarity kernel. By analogy to VSA, we call this new function encoding and computing framework Vector Function Architecture (VFA). In VFAs, vectors can represent individual data points as well as elements of a function space (a reproducing kernel Hilbert space). The algebraic vector operations, inherited from VSA, correspond to well-defined operations in function space. Furthermore, we study a previously proposed method for encoding continuous data, fractional power encoding (FPE), which uses exponentiation of a random base vector to produce randomized representations of data points and fulfills the kernel properties for inducing a VFA. We show that the distribution from which elements 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. In particular, VFAs provide an algebraic framework for implementing large-scale kernel machines with random features, extending Rahimi and Recht, 2007. Finally, we demonstrate several applications of VFA models to problems in image recognition, density estimation and nonlinear regression. Our analyses and results suggest that VFAs constitute a powerful new framework for representing and manipulating functions in distributed neural systems, with myriad applications in artificial intelligence.