Integral Formulas for Vector Signal Tensor Products Enable Efficient SO(3)-Equivariant Neural Networks
Researchers have derived integral formulas that simplify the Vector Signal Tensor Product, a mathematical generalization of the Gaunt tensor product for anti-symmetric couplings. The work provides explicit closed-form expressions for anti-symmetric Gaunt coefficients and demonstrates up to 9× reduction in required tensor product evaluations. These results enable practical implementations of Vector Signal Tensor Products in SO(3)-equivariant neural networks, a key architecture for machine learning on 3D rotational data.
A new preprint on arXiv presents integral formulas that simplify computations involving the Vector Signal Tensor Product, a recently introduced mathematical tool that extends the Gaunt tensor product to handle anti-symmetric couplings. The authors derive explicit closed-form expressions for the anti-symmetric analogues of Gaunt coefficients and show that their approach can simulate Clebsch-Gordan tensor products using a single Vector Signal Tensor Product, reducing computational overhead by up to 9×. The work also explores how Gaunt and Vector Signal Tensor Products allow researchers to balance expressivity against runtime in neural networks, and investigates low-rank decompositions of tensor product normalizations. These theoretical advances have direct implications for implementing SO(3)-equivariant neural networks, which are increasingly important for machine learning applications involving 3D rotational symmetries.
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- arXiv cs.LGCenter
Integral Formulas for Vector Signal Tensor Products
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