Nonlinear Functions of Gaussian Random Vectors Are Subgaussian: An AI-Assisted Note

This short note presents a dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear mappings. Discovered by Gemini 3.5 Flash, this result applies to any bounded function under a well-conditioned covariance. We apply this tool to resolve an open question posed by [Redacted] on sign-quantized linear maps $Y = \text{sgn}(Wx)$. By partitioning the square matrix into rectangular blocks, we bypass the typical singularity issues of square random matrices to establish the desired concentration.