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Spatial Semantic Pointers (SSPs) have recently emerged as a powerful tool forrepresenting and transforming continuous space, with numerous applications tocognitive modelling and deep learning. Fundamental to SSPs is the notion of"similarity" between vectors representing different points in $n$-dimensionalspace -- typically the dot product or cosine similarity between vectors withrotated unit-length complex coefficients in the Fourier domain. The similaritymeasure has previously been conjectured to be a Gaussian function of Euclideandistance. Contrary to this conjecture, we derive a simple trigonometric formularelating spatial displacement to similarity, and prove that, in the case wherethe Fourier coefficients are uniform i.i.d., the expected similarity is aproduct of normalized sinc functions: $\prod_{k=1}^{n} \operatorname{sinc}\left( a_k \right)$, where $\mathbf{a} \in \mathbb{R}^n$ is the spatialdisplacement between the two $n$-dimensional points. This establishes a directlink between space and the similarity of SSPs, which in turn helps bolster auseful mathematical framework for architecting neural networks that manipulatespatial structures.

A short letter on the dot product between rotated Fourier transforms