| 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 |
| |
