Smoothness prior reveals wave patterns in star data embeddings
Inducing Fourier Geometry: A Smoothness Prior Imposes Approximate Translation Equivariance in Multimodal Stellar Embeddings
Machine Learning
Summary
This paper looks at how to make computer models learn hidden patterns in star data by using a smoothness rule based on the stars’ physical properties. The authors create a prior that encourages the learned representations to change smoothly along star parameters, leading to predictable wave-like patterns linked to translation symmetry. This approach works even when the symmetry is not exact and improves performance in tasks like star classification. Their method also applies to different types of star data, revealing underlying geometric structure where none was evident before.
What this means in practice
- •For astronomy data teams: Improve star classification accuracy by embedding stellar data with a smoothness prior that captures physical parameter structure.
- •For machine learning engineers: Design multimodal data embeddings that incorporate known continuous parameter spaces to induce helpful geometric structure.
Authors
Ilay Kamai, Alex M. Bronstein, Hagai B. Perets
Abstract
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 continuous parameter space exists, its local geometry can serve as a data-informed prior.