Positivity loss in bandlimited spectral reproduction on spheres
arXiv:2609.02695
2026
Architecture
1 ideas extracted · analyzed Sep 3, 2026
What the math gives to ML
The paper studies operators that exactly reproduce every mode through degree L while restricting outputs to bandwidth N. These constraints make positivity impossible for L>=1, but the optimal excess of the uniform operator norm above one is sharply of order (L/(N+1))^2. This gives a concrete design principle for graph or spherical spectral neural layers: preserve low-frequency structure exactly, suppress frequencies above N, and optimize the transition band to minimize amplification and sign overshoot. The transferable asset is a quantitative stability-versus-bandwidth law rather than a particular spherical implementation.
Ideas from this paper
Unverified
2026
Replace an arbitrary graph spectral filter with a constrained transition-band filter that is exactly one on the lowest L graph frequencies and zero above frequency N. Train or initialize the transition coefficients to minimize the induced infinity-norm excess above one, which controls amplification and sign overshoot while retaining low-frequency information exactly. The paper predicts that widening the transition band reduces the best achievable excess quadratically rather than linearly.
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