Algebraic Transfer for Operator-Valued Gaussian Chaoses:Oriented Schatten Profiles and Singular Wick Multipliers

What the math gives to ML

The paper provides a dimension-free way to control operator-valued Gaussian polynomial maps using the maximum Schatten norm over every oriented bipartition of their coefficient tensor. The transferable asset is not the Gaussian-chaos application itself, but the profile that simultaneously controls all input-output flattenings and therefore controls random multilinear operators independently of hidden dimension. This suggests a practical regularizer and initialization rule for tensorized or polynomial neural layers: constrain sampled cut-Schatten norms of the coefficient tensor rather than only its Frobenius or spectral norm. The resulting layer should have more predictable activation tails and improved stability as tensor order and width increase.