Inducing Fourier Geometry: A Smoothness Prior Imposes Approximate Translation Equivariance in Multimodal Stellar Embeddings

Ilay Kamai, Alex M. Bronstein, Hagai B. Perets

Learned representations often contain geometric structure induced by symmetries in the underlying data. We ask whether a known continuous parameter space can instead be used to \emph{induce} such structure when the corresponding symmetry is only approximate. We introduce a graph-Laplacian smoothness prior that encourages representations to vary smoothly over a physical parameter space without specifying a representation geometry; in the continuum limit it selects Laplacian eigenfunctions and therefore predicts the Fourier-like geometry associated with translation symmetry. On multimodal stellar representations spanning six astronomical surveys, the prior induces Toeplitz Gram structure and sinusoidal modes along stellar parameters, including in modalities whose native representations lack this geometry, while modality-specific residuals preserve structure outside the imposed coordinates. Of the exact-symmetry predictions, stationarity in a recoverable warped coordinate survives; diagnostics assuming exact quantization or asymptotic rates do not. The prior also improves downstream linear-probe performance and label efficiency, suggesting a general strategy: where a meaningful…