lemmaroutine

Block coloring changes with probability 1−θ

A nonzero shift in UAU_A flips the block color with probability 1−θ1-\theta
one-block noise rate for the latent noise formula
Lemma·lem:latent-block-noise·§3.1 The quadratic block

Statement (paper)

Fix a line AA. For independent uniform (x,y)∈X⊕B(x,y)\in X\oplus B and ζ∈UA∖{0}\zeta\in U_A\setminus\{0\}, Pr⁡[Cblk((x,y)+ζ)≠Cblk(x,y)]=1−θ. \Pr\bigl[C_{\mathrm{blk}}((x,y)+\zeta) \ne C_{\mathrm{blk}}(x,y)\bigr]=1-\theta.

In plain English

For a fixed line AA, a uniformly random point (x,y)∈X⊕B(x,y)\in X\oplus B and any nonzero shift ζ∈UA\zeta\in U_A, the block coloring CblkC_{\mathrm{blk}} differs at (x,y)+ζ(x,y)+\zeta and at (x,y)(x,y) with probability exactly 1−θ1-\theta.

Why it's here

This is the one-block noise computation: it says a nonzero perturbation along UAU_A flips the block color with the same probability 1−θ1-\theta, whatever the perturbation is. The later latent-noise lemma uses this uniform value to get the factor (1−θ)(1-\theta) per block, which gives the (1−θ)t(1-\theta)^t in its change probability.
Classified routine:
A direct probability calculation over the quadratic block construction. It uses the uniformity of (x,y)(x,y) and the structure of UAU_A, in the style of standard Fourier/linear-algebra noise computations, and relies on no earlier results.

Uses

Nothing else in the paper

Used by

lemLatent coloring change probability formula

Figures

A flow diagram of one level of the latent recursion. Three boxes sit in a top row, separated by ellipses: a left child box ($u_{AJ}\in P_{t-1}$ mapped to $JC_{t-1}(u_{AJ})$), a red-highlighted selected child box ($u_{A_*J_*}+\xi_{t-1}$ mapped to $J_*C_{t-1}(u_{A_*J_*}+\xi_{t-1})$) and a right box of other child inputs. Arrows from all three, the middle one red, point to a summation box reading "Sum over every child $(A,J)\in\mathcal I$" with noisy aggregate $(x',y')=(x,y)+J_*\Delta$. That box points down to a final box, "Noisy parent output $y'+Q(x')\in B$". The figure shows how noise in only one uniformly chosen child propagates through the embedding, summation and quadratic map to the parent output.
Figure 2diagramp. 13
One level of the latent recursion, shown under its additive noise. The input space is a product over all child indices (A,J)(A,J); only one uniformly chosen child is perturbed. Each child output is embedded in UA≤X⊕BU_A\le X\oplus B, all embedded outputs are added, and the quadratic map is applied once at the parent. Here (x,y)(x,y) is the initial aggregate, and Δ\Delta is the selected child's output difference, so the noisy aggregate is (x′,y′)=(x,y)+J∗Δ(x',y')=(x,y)+J_*\Delta. The omitted children obey the same rule. The highlighted perturbation may disappear at the parent.
What it shows
A flow diagram of one level of the latent recursion. Three boxes sit in a top row, separated by ellipses: a left child box (uAJ∈Pt−1u_{AJ}\in P_{t-1} mapped to JCt−1(uAJ)JC_{t-1}(u_{AJ})), a red-highlighted selected child box (uA∗J∗+ξt−1u_{A_*J_*}+\xi_{t-1} mapped to J∗Ct−1(uA∗J∗+ξt−1)J_*C_{t-1}(u_{A_*J_*}+\xi_{t-1})) and a right box of other child inputs. Arrows from all three, the middle one red, point to a summation box reading "Sum over every child (A,J)∈I(A,J)\in\mathcal I" with noisy aggregate (x′,y′)=(x,y)+J∗Δ(x',y')=(x,y)+J_*\Delta. That box points down to a final box, "Noisy parent output y′+Q(x′)∈By'+Q(x')\in B". The figure shows how noise in only one uniformly chosen child propagates through the embedding, summation and quadratic map to the parent output.
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SupportslemLatent coloring change probability formulalemLatent alphabet gadget with detecting noiselemBlock coloring changes with probability 1−θ

Sources

PaperOpen
Section
§3.1 The quadratic block
Page
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Label
lem:latent-block-noise
Line
sections/02-latent.tex:140
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