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

2026-05-26

AI Summary

This work investigates the sub-Gaussian concentration properties of Gaussian vectors under the sign-quantized linear mapping $Y = \text{sgn}(Wx)$, addressing an open problem posed by Simone Bombari. By integrating probabilistic concentration inequalities, Gaussian process analysis, and the theory of coordinate-wise bounded nonlinear mappings under well-conditioned covariance assumptions, the authors establish—for the first time—a dimension-independent sub-Gaussian concentration bound. This result, discovered and verified with the assistance of the AI model Gemini 3.5 Flash, transcends conventional dimension-dependent analytical frameworks and provides a rigorous theoretical guarantee for the stability of sign-quantized mappings in high-dimensional random projections.

Abstract

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 answer a question of Simone Bombari on sign-quantized linear maps $Y = \text{sgn}(Wx)$.

Problem

• subgaussianity
• quantized linear maps
• Gaussian vectors
• sign quantization
• concentration bound

Innovation

• subgaussian concentration
• dimension-independent bound
• quantized linear maps
• nonlinear mappings
• Gaussian vectors