Manuscript Oai119-A@1 · Oai119-A.tex · section · fdff7292

Uniform cube and symmetric product noise

Stage 0: objects, normalizations, and claim status › Uniform cube and symmetric product noise · lines 137–171

Oai119-A@1#eq:kernel

Let

Ω_{n} = {- 1, 1}^{n}, E u = 2^{- n} x \in Ω_{n} \sum u (x) .

For $- 1 \leq ρ \leq 1$ , define

K_{ρ} (x, y) = i = 1 \prod^{n} \frac{1 + ρ x _{i} y _{i}}{2}, T_{ρ} u (y) = x \in Ω_{n} \sum K_{ρ} (y, x) u (x) .

The matrix $K_{ρ}$ is symmetric and stochastic. In probabilistic terms, $X$ is uniform and

Y_{i} = X_{i} Z_{i}, P (Z_{i} = 1) = \frac{1 + ρ}{2},

where the $Z_{i}$ are mutually independent and independent of $X$ . Both $X$ and $Y$ are uniform. Symmetry gives

T_{ρ} u (Y) = E [u (X) ∣ Y] .

These are the standard cube-noise conventions ODonnell,CourtadeKumar2014.

A Boolean function takes values in ${- 1, 1}$ . A soft summary takes values in $[- 1, 1]$ . A signed coordinate is $f (x) = σ x_{i}$ , with $σ \in {- 1, 1}$ .

Displayed formulas in this passage

Oai119-A@1#eq:kernel · line 145
$K_{ρ} (x, y) = i = 1 \prod^{n} \frac{1 + ρ x _{i} y _{i}}{2}, T_{ρ} u (y) = x \in Ω_{n} \sum K_{ρ} (y, x) u (x) .$
unlabeled · line 140
$Ω_{n} = {- 1, 1}^{n}, E u = 2^{- n} x \in Ω_{n} \sum u (x) .$
unlabeled · line 155
$Y_{i} = X_{i} Z_{i}, P (Z_{i} = 1) = \frac{1 + ρ}{2},$
unlabeled · line 161
$T_{ρ} u (Y) = E [u (X) ∣ Y] .$