An O(L^3) algorithm computes contracted Clebsch-Gordan tensor products for equivariant ML potentials using a structured angular grid and spherical Poisson bracket to handle parity-odd terms at fixed CP rank.
Spherical CNNs
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abstract
Convolutional Neural Networks (CNNs) have become the method of choice for learning problems involving 2D planar images. However, a number of problems of recent interest have created a demand for models that can analyze spherical images. Examples include omnidirectional vision for drones, robots, and autonomous cars, molecular regression problems, and global weather and climate modelling. A naive application of convolutional networks to a planar projection of the spherical signal is destined to fail, because the space-varying distortions introduced by such a projection will make translational weight sharing ineffective. In this paper we introduce the building blocks for constructing spherical CNNs. We propose a definition for the spherical cross-correlation that is both expressive and rotation-equivariant. The spherical correlation satisfies a generalized Fourier theorem, which allows us to compute it efficiently using a generalized (non-commutative) Fast Fourier Transform (FFT) algorithm. We demonstrate the computational efficiency, numerical accuracy, and effectiveness of spherical CNNs applied to 3D model recognition and atomization energy regression.
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Mapped convolutions generalize standard convolutions by decoupling sampling and weighting, enabling direct convolution on spherical and mesh data with a 17% improvement in spherical depth estimation.
SurReal architecture applies weighted Fréchet mean convolution and distance-based FC layers to complex data, improving accuracy on MSTAR (94% to 98%) and RadioML with 8-10% of baseline model size.
KamLAND finds 7 events consistent with background in 9+ kton-year exposure using NN classification and sets 90% CL DSNB flux upper limits of 38-43 cm^{-2} s^{-1} plus model-independent antineutrino flux bounds below 13.3 MeV.
GSNO uses position-dependent spherical Green's functions to create flexible neural operators that adapt to non-equivariant systems on spheres while keeping spectral efficiency and grid invariance.
TetraSphere integrates a TetraTransform based on steerable spherical neurons into VN-DGCNN to produce an O(3)-equivariant descriptor that reports new SOTA results on rotated ScanObjectNN, ModelNet40 classification, and ShapeNet segmentation.
Spherical CNNs with deformation-augmented training data achieve faster and more accurate cortical parcellation than multi-atlas or naive U-Net methods on 427 adult brains.
Sphere-Depth benchmark shows substantial performance degradation in both general and spherical-aware depth estimation models under simulated camera pose variations.
Geometric deep learning provides a unified mathematical framework based on grids, groups, graphs, geodesics, and gauges to explain and extend neural network architectures by incorporating physical regularities.
Survey organizing panoramic scene analysis literature by architectural design and training paradigm, identifying the absence of methods achieving both strict spherical equivariance and full reuse of perspective-pretrained weights, plus five evaluation protocol gaps and a six-point roadmap.
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