On the Subgaussianity of Quantized Linear Maps: An AI-Assisted Note

Simone Bombari asked us whether the 1-bit quantized random vector Y = sgn(Wx) has subgaussian norm bounded by a universal constant. Here W is an n n random Gaussian matrix, and x is an independent standard normal random vector in Rn. The question is nontrivial since the coordinates of Y are not independent. We give a strong positive answer to this question - for any bounded map instead of sgn() - using AI: AIDiscovery and Generalization (Theorem 1): To handle coordinate dependence, Gemini 3.5 Flash1 proposed decomposing the Gaussian vector into independent parts, using one part to "smooth" the sign function, and then applying Gaussian concentration for Lipschitz functions.