Complete Symbols of Equivariant Pseudodifferential Operators on Noncompact Symmetric Spaces
arXiv:2608.20313
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops a complete-symbol calculus for equivariant pseudodifferential operators on noncompact symmetric spaces, replacing the ordinary spatial-frequency symbol with a Harish-Chandra symbol on the spherical tempered dual. Its transferable asset is the combination of group equivariance, spectral localization, and dyadic symbol estimates that control how localized frequency components contribute to spatially localized kernels. A neural implementation can turn this into a multiscale equivariant spectral layer whose learnable multipliers are constrained by smoothness and decay bounds. This is most promising for data on homogeneous spaces or structured graphs, rather than as a generic replacement for Euclidean convolutions.
Ideas from this paper
Unverified
2026
Construct a neural layer as a sum of equivariant spectral operators at dyadic frequency scales, with each scale represented by a smooth learnable multiplier instead of an unconstrained dense spectral table. Enforce derivative and off-diagonal decay constraints so high-frequency components cannot create arbitrarily large or spatially nonlocal responses. On a discretized homogeneous space, this gives a multiresolution equivariant alternative to a generic graph filter or convolution kernel.
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