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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Computability.MinUncut.Graphs.EdgeMean
import OAI.Computability.MinUncut.Analysis.GaussianRows
namespace OAI
noncomputable section
open scoped BigOperators
namespace MinUncut.Composition
open MinUncut.Inner RowNoise GaussianHermite MeasureTheory
attribute [local instance] Classical.propDecidable BinaryFourier.dualFintype
attribute [local irreducible] MinUncut.Inner.gradient
variable {V W A B : Type*}
[AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A]
[AddCommGroup W] [Module F₂ W] [AddTorsor W B] [Fintype B] {m n : ℕ}
lemma comparisonRow_distance_integrable (π : A →ᵃ[F₂] B) (f : FoldedProof A) (g : FoldedProof B)
{σ : ℝ} (hσ : σ≠0) (η : ℝ) (x : Point m n) (r : Row m n)
(D : Rest (W := Forms B) r) :
Integrable (fun c => rowDistance π
(comparisonRow f σ η c x r (fun t => formPullback π (D t)))
(comparisonRow g σ η c x r D)) (gauss (Point m n)) := by
apply integrable_expect
intro z
exact ((gradient_memLp g _ hσ η x).sub (gradient_memLp f _ hσ η x)).integrable_sq
omit [Fintype A] in
lemma comparisonRow_distance_mean (π : A →ᵃ[F₂] B) (f : FoldedProof A) (g : FoldedProof B)
(σ η : ℝ) (c : Point m n → ℝ) (x : Point m n) (r : Row m n) :
(𝔼 D : Rest (W := Forms B) r, rowDistance π
(comparisonRow f σ η c x r (fun t => formPullback π (D t)))
(comparisonRow g σ η c x r D)) =
𝔼 C : FaceArray B m n,
(MinUncut.Inner.gradient f (pullbackFaces π C) σ η c x-MinUncut.Inner.gradient g C σ η c x)^2 := by
simpa only [rowDistance,sub_sq_comm] using comparisonRow_average π f g σ η c x r
lemma comparisonRow_distance_integral (π : A →ᵃ[F₂] B) (f : FoldedProof A) (g : FoldedProof B)
{σ : ℝ} (hσ : σ≠0) (η : ℝ) (x : Point m n) (r : Row m n) :
(𝔼 D : Rest (W := Forms B) r, ∫ c, rowDistance π
(comparisonRow f σ η c x r (fun t => formPullback π (D t)))
(comparisonRow g σ η c x r D) ∂gauss (Point m n)) =
jointEnergy (fun C c => MinUncut.Inner.gradient f (pullbackFaces π C) σ η c x-
MinUncut.Inner.gradient g C σ η c x) := by
rw [← integral_expect _ (fun D => comparisonRow_distance_integrable π f g hσ η x r D)]
simp_rw [comparisonRow_distance_mean]
exact integral_expect _ (fun C => ((gradient_memLp f _ hσ η x).sub
(gradient_memLp g C hσ η x)).integrable_sq)
lemma comparisonRow_distance_total (π : A →ᵃ[F₂] B) (f : FoldedProof A) (g : FoldedProof B)
(hm : 0 < m) {σ : ℝ} (hσ : σ≠0) (η : ℝ) :
(𝔼 x : Point m n, 𝔼 i : Fin m,
𝔼 D : Rest (W := Forms B) (⟨i,face x i⟩ : Row m n), ∫ c, rowDistance π
(comparisonRow f σ η c x ⟨i,face x i⟩ (fun t => formPullback π (D t)))
(comparisonRow g σ η c x ⟨i,face x i⟩ D) ∂gauss (Point m n)) =
averagedEnergy (m := m) (n := n) (fun C c x => MinUncut.Inner.gradient f (pullbackFaces π C) σ η c x-
MinUncut.Inner.gradient g C σ η c x) := by
let : NeZero m := ⟨hm.ne'⟩
simp_rw [comparisonRow_distance_integral π f g hσ,Fintype.expect_const]
exact (averagedEnergy_eq_expect_joint _ (fun C x => (gradient_memLp f _ hσ η x).sub
(gradient_memLp g C hσ η x))).symm
end MinUncut.Composition
open scoped BigOperators
namespace MinUncut.Outer
open MinUncut.Inner OuterSmoothness MinUncut.Composition RowNoise GaussianHermite MeasureTheory
attribute [local instance] Classical.propDecidable BinaryFourier.dualFintype
attribute [local irreducible] MinUncut.Inner.gradient
variable {Name I S : Type*} [Fintype I] [Fintype S] {m n : ℕ}
def listTerm (M u θ : ℝ) (q k : ℕ) : ℝ :=
(⌈M^2/u^2⌉₊ : ℝ)*(⌈M^2/θ^2⌉₊ : ℝ)*hintRate q^k
def remainderTerm (M u θ : ℝ) (q k t : ℕ) : ℝ :=
(256/θ^4)*(u^2*M^2+Real.sqrt (((2 : ℝ)^(4*(q+3)*k)-1)*(k : ℝ)^2/t)*(M+M^2/u)^4)
def ProofFamily.atomAt (f : ProofFamily Name I) (σ η θ : ℝ) (x : Point m n) (i : Fin m) :
EdgeStatistic Name I :=
fun U h pos => rowEventProbability (f.second (secondQuestion U h pos)) σ η θ x i
def ProofFamily.distanceAt (f : ProofFamily Name I) (σ η : ℝ) (x : Point m n) (i : Fin m) :
EdgeStatistic Name I := fun U h pos =>
𝔼 D : Rest (W := Forms (SecondAlphabet (secondQuestion U h pos))) (⟨i,face x i⟩ : Row m n),
∫ c, (f.rows σ η c x ⟨i,face x i⟩).distance U h pos D ∂gauss (Point m n)
lemma ProofFamily.atomAt_integral (f : ProofFamily Name I) (equations : S → Equation Name)
(k : ℕ) {σ θ : ℝ} (hσ : σ≠0) (hθ : 0<θ) (η : ℝ)
(x : Point m n) (i : Fin m) :
(∫ c, backgroundMean equations k ((f.rows σ η c x ⟨i,face x i⟩).atom θ) ∂gauss (Point m n)) =
edgeMean equations k (f.atomAt σ η θ x i) := by
rw [integral_backgroundMean _ _ _ (f.rows_atom_integrable hσ hθ η x i)]
unfold backgroundMean edgeMean ProofFamily.atomAt rowEventProbability LocalRows.atom ProofFamily.rows
simp_rw [← rowAtomIndicator_eq_list _ σ η hθ]
rfl
lemma ProofFamily.distanceAt_integral (f : ProofFamily Name I) (equations : S → Equation Name)
(k : ℕ) {σ : ℝ} (hσ : σ≠0) (η : ℝ) (x : Point m n) (i : Fin m) :
(∫ c, backgroundMean equations k (f.rows σ η c x ⟨i,face x i⟩).distance ∂gauss (Point m n)) =
edgeMean equations k (f.distanceAt σ η x i) :=
integral_backgroundMean _ _ _ (f.rows_distance_integrable hσ η x _)
theorem ProofFamily.row_soundness [Fintype Name] [Nonempty I] [Nonempty S]
(f : ProofFamily Name I) (equations : S → Equation Name)
(hsound : ∀ s : Name → F₂, equationFraction equations s≤3/4)
{k : ℕ} (hk : k≤Fintype.card I) {σ u θ : ℝ} (hσ : 0<σ) (hu : 0<u) (hθ : 0<θ)
(η : ℝ) (x : Point m n) (i : Fin m) :
edgeMean equations k (f.atomAt σ η θ x i) ≤
listTerm (Real.sqrt (n^m:ℕ)/σ) u θ (Fintype.card (Row m n)-1) k +
(16/θ^2)*edgeMean equations k (f.distanceAt σ η x i) +
remainderTerm (Real.sqrt (n^m:ℕ)/σ) u θ (Fintype.card (Row m n)-1) k (Fintype.card I) := by
let M : ℝ := Real.sqrt (n^m:ℕ)/σ
let L := listTerm M u θ (Fintype.card (Row m n)-1) k
let R := remainderTerm M u θ (Fintype.card (Row m n)-1) k (Fintype.card I)
have hpoint (c : Point m n → ℝ) :
backgroundMean equations k ((f.rows σ η c x ⟨i,face x i⟩).atom θ) ≤
L+(16/θ^2)*backgroundMean equations k (f.rows σ η c x ⟨i,face x i⟩).distance+R := by
have h := (f.rows σ η c x ⟨i,face x i⟩).atom_bound equations hsound hk
(by positivity : 0≤M) hu hθ (f.rows_first_bound hσ η c x _) (f.rows_second_bound hσ η c x _)
simpa only [rest_card,L,R,listTerm,remainderTerm] using h
have hiA := backgroundMean_integrable equations k _ (f.rows_atom_integrable hσ.ne' hθ η x i)
have hiD := backgroundMean_integrable equations k _ (f.rows_distance_integrable hσ.ne' η x ⟨i,face x i⟩)
have h := integral_mono hiA (((integrable_const L).add (hiD.const_mul (16/θ^2))).add (integrable_const R)) hpoint
rw [f.atomAt_integral equations k hσ.ne' hθ η x i] at h
simp only [Pi.add_apply] at h
have hiL : Integrable (fun c => L+(16/θ^2)*backgroundMean equations k (f.rows σ η c x ⟨i,face x i⟩).distance) (gauss (Point m n)) :=
(integrable_const L).add (hiD.const_mul _)
rw [integral_add hiL (integrable_const R), integral_add (integrable_const L) (hiD.const_mul (16/θ^2)),integral_const_mul, f.distanceAt_integral equations k hσ.ne' η x i] at h
simpa only [integral_const,MeasureTheory.probReal_univ,one_smul] using h
end MinUncut.Outer
end
end OAI
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