## Abstract

Hyperdimensional computing combines very high-dimensional vector spaces (e.g. 10,000 dimensional) with a set of carefully designed operators to perform symbolic computations with large numerical vectors. The goal is to exploit their representational power and noise robustness for a broad range of computational tasks. Although there are surprising and impressive results in the literature, the application to practical problems in the area of robotics is so far very limited. In this work, we aim at providing an easy to access introduction to the underlying mathematical concepts and describe the existing computational implementations in form of vector symbolic architectures (VSAs). This is accompanied by references to existing applications of VSAs in the literature. To bridge the gap to practical applications, we describe and experimentally demonstrate the application of VSAs to three different robotic tasks: viewpoint invariant object recognition, place recognition and learning of simple reactive behaviors. The paper closes with a discussion of current limitations and open questions.

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## Notes

n-sphere: a hypersphere in the (n+1)-dimensional space. n-cap: portion of an n-sphere cut off by a hyperplane.

Please keep in mind, that the unit of the surface area of an n-sphere is an n-dimensional object, thus the unit along the vertical axis changes and the values along the curves are not directly comparable. Nevertheless, the fact that there is a local maximum of the surface area of the almost orthogonal range is surprising. However, it is a direct consequence of the local maximum of the surface area of the whole unit n-sphere (which in turn becomes intuitive based on the recursive expression of the surface area \(A_{n+1} = A_n \cdot \frac{n}{2\pi }\) since for \(n>2\pi\) this factor becomes smaller one).

Details: the red curve in the left plot evaluates vector similarities (the query image index

*q*is known and we compare the similarity of \(I^k_x + I^k_y\) and \(I^{q=k}_z\)), the red curve in the right plot evaluates the accuracy of a nearest neighbor query (the query image index*q*is not known to the system and it returns the index*k*of the nearest neighbor to \(I^q_z\) of all \(I^k_x + I^k_y\), \(k\in \{1 \ldots 1000\}\)).*x*is fixed at viewing angle \(0^{\circ }\).*y*varies from \(0^{\circ }\) to \(350^{\circ }\). The horizontal axis is the mean angular distance from*z*to*x*and*y*. As a reading example: in the left plot, the red curve evaluated at \(90^{\circ }\) means that for \(x=0^{\circ }\), \(y=180^{\circ }\), \(z=90^{\circ }\) (e.g. the images from Fig. 6), the average cosine distance of the bundle \((I^k_0 + I^k_{180})\) and \(I^k_{90}\) is about 0.17, and the right plot tells us that for about 53% of the objects the query image was most similar to the correct bundle. For comparison without bundling, the blue curves in Fig. 7 show the results when comparing the query image to the individual images \(I^k_x\) and \(I^k_y\) (instead of their bundle). For the distance evaluation in the left plot, we use the*closest*of the two individual results for each query. For the query results in the right plot, all views \(I^k_x\) and \(I^k_y\) are stored in the database and a single query is made (the number of data base entries and thus comparisons has now doubled compared to the bundling approach). The VSA approach not only reduces the number of comparison, it also performs slightly better than using individual comparisons in both plots.This work was previously presented at an IROS workshop, see [26] for details.

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Neubert, P., Schubert, S. & Protzel, P. An Introduction to Hyperdimensional Computing for Robotics.
*Künstl Intell* **33**, 319–330 (2019). https://doi.org/10.1007/s13218-019-00623-z

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DOI: https://doi.org/10.1007/s13218-019-00623-z