Section 8 global variance: operator families #
Positivity and normalization lemmas for the point-conditioned outcome operator families used to build the global-variance comparisons.
References #
references/ldt-paper/expansion.texblueprint/src/chapter/ch06_variance.tex
The squared norm expression controlled by lem:generalize-b for a fixed g.
Uses bipartite state ψbi on d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The polynomial-averaged deviation controlled by lem:generalize-b.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The line-collision residual obtained after expanding the projective line
measurement in lem:generalize-b and moving the polynomial weight from
(G_g)^{1/2} to G_g. The remaining unproved analytic step is to bound this
quantity by Schwartz--Zippel and the submeasurement property of G.
Paper origin: references/ldt-paper/expansion.tex:273-288
(\label{lem:generalize-b}).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Uniform line/parameter seed expansion of the lem:generalize-b collision residual.
This is the paper's expansion.tex, lines 286--288, after replacing an
incident line question (ℓ,u) by a line ℓ and affine parameter t with
u = ℓ(t), but before commuting the finite average over t past the finite
sum over line answers f. The equality from the original collision residual to
this seed expansion is proved in
generalizeBCollisionResidual_eq_seedCollisionExpansion.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Explicit line/parameter expansion of the lem:generalize-b collision residual.
This is the commuted finite-sum form of generalizeBSeedCollisionExpansion: for
fixed ℓ and f, the coefficient is the fraction of parameters where the line
answer f both collides with g|_ℓ at t and is not equal to g|_ℓ.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Aggregated family for the left-hand side of lem:generalize-b
on the bipartite space d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Aggregated family for the right-hand side of lem:generalize-b
on the bipartite space d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Aggregated family for A^u_[g(u)] ⊗ (G_g)^{1/2}
on the bipartite space d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Aggregated family for A^v_[g(v)] ⊗ (G_g)^{1/2}
on the bipartite space d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The same weighted operator on the first independently sampled point.
On the bipartite space d * d.
Equations
- MIPStarRE.LDT.GlobalVariance.globalVarianceLeftFamily params strategy G = MIPStarRE.LDT.GlobalVariance.localVarianceLeftFamily params strategy G
Instances For
The same weighted operator on the second independently sampled point.
On the bipartite space d * d.
Equations
- MIPStarRE.LDT.GlobalVariance.globalVarianceRightFamily params strategy G = MIPStarRE.LDT.GlobalVariance.localVarianceRightFamily params strategy G
Instances For
The edgewise squared norm expression in lem:local-variance-of-points.
Uses bipartite state ψbi on d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The independently sampled squared norm expression in lem:global-variance-of-points.
Uses bipartite state ψbi on d * d.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The polynomial-averaged local squared norm expression.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The displayed error term in lem:generalize-b.
Instances For
The post-triangle-inequality error term for the six-step transport in
lem:local-variance-of-points.
The six comparisons at references/ldt-paper/expansion.tex, lines 305--311
have errors 2δ, 2ε, md/q, md/q, 2ε, and 2δ, whose sum is
4ε + 4δ + 2md/q. The paper then applies
prop:triangle-inequality-for-approx_delta (preliminaries.tex:622--642) with
k = 6, yielding 6 * (4ε + 4δ + 2md/q). This is the genuine residual that
the six-step chain is expected to prove before it is absorbed by the displayed
24(ε + δ + md/q) slack.
Equations
- MIPStarRE.LDT.GlobalVariance.localVarianceTransportChainError params eps delta = 6 * (4 * eps + 4 * delta + 2 * MIPStarRE.LDT.GlobalVariance.generalizeBError params)
Instances For
The displayed error term in lem:local-variance-of-points.
Equations
- MIPStarRE.LDT.GlobalVariance.localVarianceOfPointsError params eps delta = 24 * (eps + delta + MIPStarRE.LDT.GlobalVariance.generalizeBError params)
Instances For
The displayed error term in lem:global-variance-of-points.
Equations
- MIPStarRE.LDT.GlobalVariance.globalVarianceOfPointsError params eps delta = 24 * ↑params.m * (eps + delta + MIPStarRE.LDT.GlobalVariance.generalizeBError params)