On periodic distributed representations using Fourier embeddings

Coherent, continuous spatial representations are critical for synthesizing physical and perceptual phenomena into a single representational space. Radial basis kernels provide a path forward for this type of distributed representation. In this work, we aim to characterize and analyze common radial basis kernels realizable in the neurally-plausible framework of spatial semantic pointers. Further, we analyze previous radial basis kernel work based on grid cell-like representations and demonstrate that such representations are both capable of and optimal for realizing radial basis kernels.

The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=Ω((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}δ)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilonδ))))$, RFFs suffice, with probability $1-δ$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.

Numbers have algebraic structure that standard neural embeddings often fail to expose. We introduce Prime Fourier Embeddings (PFE), which encode integers as prime-indexed (cos, sin) pairs derived from the harmonic analysis of Q, providing a pre-structured representation in which modular arithmetic reduces to selecting the relevant prime channel rather than discovering algebraic structure from scratch. We prove that any linear map equivariant with respect to the product group action on PFE must be block-diagonal with one independent block per prime -- a consequence of Schur's lemma applied to the resulting character decomposition. For square-free composite moduli, the Chinese Remainder Theorem predicts which prime channels are task-relevant. Both predictions are confirmed empirically: ablation studies show specialization ratios exceeding 500x between task-relevant and task-irrelevant channels, with perfect in-distribution test accuracy across all square-free composite moduli tested.