A note on a sparse sampling conjecture
arXiv:2608.29217
2026
Geometry
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper proves an exact identifiability theorem for sets from Fourier samples taken on a dual lattice. For connected finite unions of convex bodies satisfying an alias-free sparse-lattice condition, equality of all sampled indicator transforms determines the set up to a lattice translation. The transferable asset is a mathematically explicit condition preventing periodic aliasing in Fourier-feature representations. This suggests combining finite Fourier supervision for neural implicit shapes with a sparse-lattice regularizer or lattice-selection procedure.
Ideas from this paper
Unverified
2026
Train a neural implicit occupancy or signed-distance model with Fourier coefficients sampled on a dual lattice, while explicitly preventing spatial aliasing under the corresponding periodic lattice. The spatial reconstruction loss is supplemented by a finite Fourier loss and a penalty for shape-point differences that approach nonzero lattice vectors.
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