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import OAI.Computability.VertexCover.Analysis.OwnTotalEnergyNo
import OAI.Computability.VertexCover.Analysis.Poincare
namespace OAI
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namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory
theorem ownMean_poincare (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(j : J) (i : Φ.Query d) :
variance (Φ.ownMean J frozen c0 A hA j i)
(VertexCover.Cube.law (Fin (Φ.WeightDimension d))) ≤
2 * ∫ s, ∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2
∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)) := by
apply VertexCover.Cube.poincare (Φ.ownMean_lipschitz J frozen c0 A hA j i)
(Φ.ownGradient J frozen c0 A hA j i)
(Φ.ownGradient_measurable J frozen c0 A hA j i)
· intro s k
exact VertexCover.Threshold.coordinate_bound (Φ.ownGradient_block_bound J frozen c0 A hA j i s) k
· exact Φ.ownMean_partial_ae J frozen c0 A hA j i
noncomputable def ownTotalVariance (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : ℝ :=
∑ j : J, VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J =>
variance (Φ.ownMean J frozen c0 A hA j (Φ.query (Φ.spliceSeeds J frozen hidden) j))
(VertexCover.Cube.law (Fin (Φ.WeightDimension d))))
theorem ownGradient_sq_integrable (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(j : J) (i : Φ.Query d) :
Integrable (fun s => ∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2)
(VertexCover.Cube.law (Fin (Φ.WeightDimension d))) := by
apply integrable_finsetSum
intro k hk
apply Integrable.of_bound ((Φ.ownGradient_measurable J frozen c0 A hA j i k).pow_const 2).aestronglyMeasurable 1
filter_upwards [] with s
have hb := VertexCover.Threshold.coordinate_bound (Φ.ownGradient_block_bound J frozen c0 A hA j i s) k
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
nlinarith [abs_nonneg (Φ.ownGradient J frozen c0 A hA j i s k),
sq_abs (Φ.ownGradient J frozen c0 A hA j i s k)]
theorem ownTotalVariance_le_energy (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
Φ.ownTotalVariance J frozen c0 A hA ≤ 2 * Φ.ownTotalEnergy J frozen c0 A hA := by
classical
unfold ownTotalVariance ownTotalEnergy
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro j hj
rw [VertexCover.integral_finiteMean _ _ (fun hidden =>
Φ.ownGradient_sq_integrable J frozen c0 A hA j (Φ.query (Φ.spliceSeeds J frozen hidden) j))]
calc
_ ≤ VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J =>
2 * ∫ s, ∑ k, (Φ.ownGradient J frozen c0 A hA j
(Φ.query (Φ.spliceSeeds J frozen hidden) j) s k)^2
∂VertexCover.Cube.law (Fin (Φ.WeightDimension d))) :=
VertexCover.finiteMean_mono (fun hidden => Φ.ownMean_poincare J frozen c0 A hA j _)
_ = _ := by simp only [VertexCover.finiteMean, ← Finset.mul_sum]; ring
theorem ownTotalVariance_no_upper (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(hval : Φ.value ≤ Parameters.σ m)
(J : Finset (Fin (Parameters.d m))) (hJ : J.card = Parameters.h m)
(frozen : Φ.Seeds (Parameters.d m)) (c0 : Φ.Coordinate (Parameters.d m) → ℝ)
(A : Finset (Φ.Coordinate (Parameters.d m) → ℝ)) (hA : A.Nonempty) :
Φ.ownTotalVariance J frozen c0 A hA < 7 := by
have he := Φ.ownTotalVariance_le_energy J frozen c0 A hA
have hu := Φ.ownTotalEnergy_no_upper m hm hval J hJ frozen c0 A hA
linarith
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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