Boolean Noise Hypercontractivity
See how product-space noise turns a second-moment bound into a fourth-moment bound.
Free theory articleSee how product-space noise turns a second-moment bound into a fourth-moment bound.
Concept & interpretation
Let's calculate the L2 and L4 norms for the Product function f(x, y) = (1 + a × x)(1 + a × y) with Amplitude a = 0.5 and Noise retention ρ = Threshold (1 / √3).
Experiment
Observe how the hypercontractive inequality behaves across different noise levels using the module.
- Set Function to One bit and Amplitude to 0.5.
- Set Noise retention to Threshold. The Norm ratio is ≈ 0.991, and the Universal guarantee is Yes.
- Change Noise retention to 1. The Norm ratio is now ≈ 1.132. The L4 norm exceeds the L2 norm, showing that hypercontractivity fails without sufficient noise damping.
- Set Function to Product and test Noise retention at Threshold and 3/4 to see that the guarantee only holds globally for ρ ≤ 1/√3.