Boolean Noise Hypercontractivity

Worked Example

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).

The four corners of the boolean hypercube are (1, 1), (1, -1), (-1, 1), and (-1, -1). The original function is f(x, y) = (1 + 0.5x)(1 + 0.5y). The noisy function Tf is Tf(x, y) = 1 + a × ρ × x + a × ρ × y + a × a × ρ × ρ × x × y.

Calculation for each corner (x, y):
1. (1, 1):
   f = (1 + 0.5)(1 + 0.5) = 1.5 * 1.5 = 2.25
   f² = 5.0625

   Tf = 1 + 0.5 * (1/√3) * 1 + 0.5 * (1/√3) * 1 + 0.25 * (1/3) * 1
      = 1 + 1/√3 + 1/12 ≈ 1 + 0.57735 + 0.08333 = 1.66068
   (Tf)⁴ = (1.66068)⁴ ≈ 7.6067

2. (1, -1):
   f = (1 + 0.5)(1 - 0.5) = 1.5 * 0.5 = 0.75
   f² = 0.5625

   Tf = 1 + 0.5 * (1/√3) * 1 + 0.5 * (1/√3) * (-1) + 0.25 * (1/3) * (-1)
      = 1 + 0.28867 - 0.28867 - 0.08333 = 0.91667
   (Tf)⁴ = (0.91667)⁴ ≈ 0.7061

3. (-1, 1):
   By symmetry, same as (1, -1)
   f² = 0.5625
   (Tf)⁴ ≈ 0.7061

4. (-1, -1):
   f = (1 - 0.5)(1 - 0.5) = 0.5 * 0.5 = 0.25
   f² = 0.0625

   Tf = 1 - 0.28867 - 0.28867 + 0.08333 = 0.5060
   (Tf)⁴ = (0.5060)⁴ ≈ 0.0655

Expected Norms:
L2_sum = 5.0625 + 0.5625 + 0.5625 + 0.0625 = 6.25
L2 = √(6.25 / 4) = √1.5625 = 1.25

L4_sum ≈ 7.6067 + 0.7061 + 0.7061 + 0.0655 ≈ 9.0844
L4 = (9.0844 / 4)^(1/4) = (2.2711)^(1/4) ≈ 1.227576655221353

Thus, L2 = 1.25 and L4 ≈ 1.227. Since L4 ≤ L2 (norm ratio ≈ 0.982), Bonami hypercontractivity is satisfied at the threshold.

Experiment

Observe how the hypercontractive inequality behaves across different noise levels using the module.

1. Set Function to One bit and Amplitude to 0.5.
2. Set Noise retention to Threshold. The Norm ratio is ≈ 0.991, and the Universal guarantee is Yes.
3. 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.
4. 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.

Limitations

The interactive visualization here only computes the norm values over the 2-dimensional boolean hypercube (four corners). While Bonami's lemma applies universally to polynomials on n-dimensional boolean hypercubes with arbitrary n, computing norms explicitly over 2ⁿ points becomes computationally intractable for large n, requiring analytical methods rather than direct summation. Additionally, floating-point roundoff restricts the numerical precision when tracking exactly zero deviations or differences.

References

This behavior directly reflects Bonami's Hypercontractivity theorem.

• Ryan O'Donnell, "Analysis of Boolean Functions". See Lecture 16 for a full derivation of the Two-Point Inequality and Bonami's Lemma.
  https://www.cs.cmu.edu/~odonnell/boolean-analysis/lecture16.pdf