Spherical Harmonics
A set of orthogonal basis functions defined on the sphere, characterized by degree l representing angular momentum features. They are a core mathematical tool for constructing SO(3)-equivariant neural networks such as Tensor Field Networks.
Timeline (last 90 days)
For a set of residual isotropic spatial sources at fixed density, positive gain can be achieved on every fixed finite-order spherical harmonic.
Complete-class certificates can distinguish SH0, SH1 and SH2 fits on a learned density.
Spherical harmonics characterize angular momentum by order l, where l=0 corresponds to scalars, l=1 corresponds to vectors, and higher orders correspond to tensor representations.
Legendre polynomials are a set of orthogonal polynomials defined on the interval [-1, 1], which generalize to spherical harmonics when extended to spherical coordinates.
When the activation function is ReLU, the target function is a pure spherical harmonic, and under Gaussian universality assumptions, the student's error scales approximately as the square of the teacher's error
The paper analyzes two extended settings: more general stopping times and target functions supported on multiple spherical harmonic orders
All Facts (6)
For a set of residual isotropic spatial sources at fixed density, positive gain can be achieved on every fixed finite-order spherical harmonic.
50%UnverifiedComplete-class certificates can distinguish SH0, SH1 and SH2 fits on a learned density.
50%UnverifiedSpherical harmonics characterize angular momentum by order l, where l=0 corresponds to scalars, l=1 corresponds to vectors, and higher orders correspond to tensor representations.
50%UnverifiedLegendre polynomials are a set of orthogonal polynomials defined on the interval [-1, 1], which generalize to spherical harmonics when extended to spherical coordinates.
50%UnverifiedWhen the activation function is ReLU, the target function is a pure spherical harmonic, and under Gaussian universality assumptions, the student's error scales approximately as the square of the teacher's error
50%UnverifiedThe paper analyzes two extended settings: more general stopping times and target functions supported on multiple spherical harmonic orders
50%