| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.ExpectResampleRow |
| import OAI.Computability.MinUncut.Analysis.LowGradient |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Inner |
| open MeasureTheory ProbabilityTheory GaussianHermite RowNoise |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| attribute [local irreducible] query pullQuery thirdFailure |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
| {m n : ℕ} |
|
|
| lemma spatialEnergy_eq_expect (v : Point m n → ℝ) : |
| spatialEnergy v = 𝔼 x : Point m n, (v x)^2 := by |
| simp [spatialEnergy,Fintype.expect_eq_sum_div_card,Point,div_eq_mul_inv,mul_comm] |
|
|
| lemma spatial_young (v w : Point m n → ℝ) {a : ℝ} (ha : 0<a) : |
| (𝔼 x : Point m n, v x*w x) ≤ a*spatialEnergy v+(4*a)⁻¹*spatialEnergy w := by |
| rw [spatialEnergy_eq_expect,spatialEnergy_eq_expect] |
| simp only [Finset.mul_expect,← Finset.expect_add_distrib] |
| apply Finset.expect_le_expect |
| intro x _ |
| have he : (4*a)*((4*a)⁻¹) = 1 := mul_inv_cancel₀ (by positivity) |
| nlinarith [sq_nonneg (2*a*v x-w x)] |
|
|
| lemma gradient_correlation_truncate (f : FoldedProof A) (B : FaceArray A m n) |
| (hn : 0<n) {σ η T a : ℝ} (hσ : σ≠0) (hη : 0<η) (hT : 0<T) (ha : 0<a) |
| (c : Point m n → ℝ) (v : Point m n → ℝ) : |
| (𝔼 x : Point m n, v x*gradient f B σ η c x) ≤ |
| T*faceCorrelation v+a*spatialEnergy v+ |
| (σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)*thirdRejection f B σ η c := by |
| have hE : 0 ≤ spatialEnergy v := by rw [spatialEnergy_eq_expect]; positivity |
| have hW := tableMass_le_thirdRejection f B σ hη c |
| have hcoeff : 0 ≤ σ⁻¹^2/(4*a*T) := by positivity |
| have hsmall : 0 ≤ (σ⁻¹^2/(4*a*T))*tableMass f B σ η c := |
| mul_nonneg hcoeff (Finset.sum_nonneg (fun _ _ => abs_nonneg _)) |
| have hmid : (𝔼 x : Point m n, v x*gradient f B σ η c x) ≤ |
| T*faceCorrelation v+a*spatialEnergy v+(σ⁻¹^2/(4*a*T))*tableMass f B σ η c := by |
| by_cases h : tableMass f B σ η c ≤ T |
| · have hz := (le_abs_self _).trans (gradient_convex_correlation f B hn hσ hη.ne' c v) |
| have hm := mul_le_mul_of_nonneg_right h (faceCorrelation_nonneg v) |
| nlinarith [mul_nonneg ha.le hE] |
| · have hy := spatial_young v (gradient f B σ η c) ha |
| have hg := mul_le_mul_of_nonneg_left (gradient_spatialEnergy_le f B hn hσ η c) |
| (show 0≤(4*a)⁻¹ by positivity) |
| have hw := mul_le_mul_of_nonneg_left (le_of_lt (lt_of_not_ge h)) hcoeff |
| have he : (σ⁻¹^2/(4*a*T))*T = (4*a)⁻¹*σ⁻¹^2 := by field_simp |
| rw [he] at hw |
| nlinarith [mul_nonneg hT.le (faceCorrelation_nonneg v)] |
| exact hmid.trans (by nlinarith [mul_le_mul_of_nonneg_left hW hcoeff]) |
|
|
| omit [Fintype A] in |
| lemma thirdRejection_measurable (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) : |
| Measurable (thirdRejection f B σ η) := by |
| have hh (z : Code m n) : Measurable |
| (fun p : ((Point m n → ℝ) × (Point m n → ℝ)) × (Code m n → ℝ) => |
| thirdFailure f B η (p.1.1+σ•p.1.2) p.2 z) := |
| (measurable_thirdFailure f B η z).comp (show Measurable |
| (fun p : ((Point m n → ℝ) × (Point m n → ℝ)) × (Code m n → ℝ) => |
| (p.1.1+σ•p.1.2,p.2)) by fun_prop) |
| have hi (z : Code m n) : Measurable (fun c : Point m n → ℝ => |
| ∫ g : Point m n → ℝ, ∫ l : Code m n → ℝ, |
| thirdFailure f B η (c+σ•g) l z ∂gauss (Code m n) ∂gauss (Point m n)) := |
| (hh z).stronglyMeasurable.integral_prod_right.measurable.stronglyMeasurable.integral_prod_right.measurable |
| unfold thirdRejection |
| simp only [Fintype.expect_eq_sum_div_card] |
| exact (Finset.measurable_sum _ (fun z _ => hi z)).div_const _ |
|
|
| omit [Fintype A] in |
| lemma thirdRejection_nonneg (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) : 0 ≤ thirdRejection f B σ η c := by |
| apply Finset.expect_nonneg |
| intro z _ |
| exact integral_nonneg (fun g => integral_nonneg (fun l => thirdFailure_nonneg f B η _ l z)) |
|
|
| omit [Fintype A] in |
| lemma thirdRejection_abs_le (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) : |thirdRejection f B σ η c| ≤ 1 := by |
| have he (z : Code m n) (g : Point m n → ℝ) : |
| |∫ l, thirdFailure f B η (c+σ•g) l z ∂gauss (Code m n)| ≤ 1 := by |
| simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using |
| norm_integral_le_of_norm_le_const (μ := gauss (Code m n)) |
| (f := fun l => thirdFailure f B η (c+σ•g) l z) (C := 1) |
| (ae_of_all _ (fun l => by simpa only [Real.norm_eq_abs] using thirdFailure_abs_le f B η _ l z)) |
| have hb (z : Code m n) : |
| |∫ g, ∫ l, thirdFailure f B η (c+σ•g) l z ∂gauss (Code m n) ∂gauss (Point m n)| ≤ 1 := by |
| simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using |
| norm_integral_le_of_norm_le_const (μ := gauss (Point m n)) |
| (f := fun g => ∫ l, thirdFailure f B η (c+σ•g) l z ∂gauss (Code m n)) (C := 1) |
| (ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using he z g)) |
| exact (Finset.abs_expect_le _ _).trans (by |
| simpa using Finset.expect_le_expect (s := Finset.univ) (fun z _ => hb z)) |
|
|
| omit [Fintype A] in |
| lemma thirdRejection_integrable (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) : |
| Integrable (thirdRejection f B σ η) (gauss (Point m n)) := |
| Integrable.of_bound (thirdRejection_measurable f B σ η).aestronglyMeasurable 1 |
| (ae_of_all _ (fun c => by simpa only [Real.norm_eq_abs] using thirdRejection_abs_le f B σ η c)) |
|
|
| lemma lowGradient_continuous (D s : ℕ) (f : FoldedProof A) (σ η : ℝ) |
| (B : FaceArray A m n) (x : Point m n) : |
| Continuous (fun c => lowGradient D s f σ η B c x) := |
| selectedProject_continuous _ _ _ |
|
|
| lemma lowGradient_correlation_continuous (D s : ℕ) (f : FoldedProof A) (σ η : ℝ) |
| (B : FaceArray A m n) : |
| Continuous (fun c => faceCorrelation (lowGradient D s f σ η B c)) := by |
| apply Continuous.finset_sup'_apply |
| intro z _ |
| simp only [Fintype.expect_eq_sum_div_card] |
| exact ((continuous_finsetSum _ (fun x _ => |
| (lowGradient_continuous D s f σ η B x).mul continuous_const)).div_const _).abs |
|
|
| lemma lowGradient_correlation_integrable (D s : ℕ) (f : FoldedProof A) (σ η : ℝ) |
| (B : FaceArray A m n) : |
| Integrable (fun c => faceCorrelation (lowGradient D s f σ η B c)) (gauss (Point m n)) := by |
| have hi : Integrable (fun c => 𝔼 x : Point m n, |lowGradient D s f σ η B c x|) |
| (gauss (Point m n)) := by |
| simp only [Fintype.expect_eq_sum_div_card] |
| exact (integrable_finsetSum _ (fun x _ => |
| ((lowGradient_memLp D s f σ η B x).integrable (by norm_num)).abs)).div_const _ |
| exact hi.mono' (lowGradient_correlation_continuous D s f σ η B).aestronglyMeasurable |
| (ae_of_all _ (fun c => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (faceCorrelation_nonneg _)] |
| exact faceCorrelation_le_l1 _)) |
|
|
| def averagedCorrelation (u : FaceArray A m n → (Point m n → ℝ) → Point m n → ℝ) : ℝ := |
| 𝔼 B, ∫ c, faceCorrelation (u B c) ∂gauss (Point m n) |
|
|
| def thirdError (f : FoldedProof A) (σ η : ℝ) : ℝ := |
| 𝔼 B : FaceArray A m n, ∫ c, thirdRejection f B σ η c ∂gauss (Point m n) |
|
|
| lemma lowGradient_inner_identity (D s : ℕ) (f : FoldedProof A) |
| {σ : ℝ} (hσ : σ≠0) (η : ℝ) : |
| (𝔼 B : FaceArray A m n, ∫ c, (𝔼 x : Point m n, |
| lowGradient D s f σ η B c x*gradient f B σ η c x) ∂gauss (Point m n)) = |
| averagedEnergy (lowGradient (m := m) (n := n) D s f σ η) := by |
| have hi (B : FaceArray A m n) : |
| (∫ c, (𝔼 x : Point m n, lowGradient D s f σ η B c x*gradient f B σ η c x) |
| ∂gauss (Point m n)) = |
| (𝔼 x : Point m n, ∫ c, lowGradient D s f σ η B c x*gradient f B σ η c x |
| ∂gauss (Point m n)) := by |
| simp only [Fintype.expect_eq_sum_div_card,integral_div] |
| rw [integral_finsetSum (f := fun x c => lowGradient D s f σ η B c x*gradient f B σ η c x) _ (fun x _ => |
| (lowGradient_memLp D s f σ η B x).integrable_mul (gradient_memLp f B hσ η x))] |
| simp_rw [hi] |
| rw [Finset.expect_comm] |
| have he (x : Point m n) : |
| (𝔼 B : FaceArray A m n, ∫ c, |
| lowGradient D s f σ η B c x*gradient f B σ η c x ∂gauss (Point m n)) = |
| jointEnergy (fun B c => lowGradient D s f σ η B c x) := by |
| change jointInner (selectedProject (rectangle D s) (fun B c => gradient f B σ η c x)) |
| (fun B c => gradient f B σ η c x) = _ |
| rw [jointInner_symm,selectedProject_source_inner _ _ (fun B => gradient_memLp f B hσ η x), |
| ← selectedProject_energy] |
| rfl |
| simp_rw [he] |
| rw [averagedEnergy_eq_jointEnergy _ (lowGradient_memLp D s f σ η)] |
| simp [Point,Fintype.expect_eq_sum_div_card,div_eq_mul_inv,mul_comm] |
|
|
| lemma lowGradient_correlation_truncated (D s : ℕ) (f : FoldedProof A) |
| (hn : 0<n) {σ η T a : ℝ} (hσ : σ≠0) (hη : 0<η) (hT : 0<T) (ha : 0<a) : |
| averagedEnergy (lowGradient (m := m) (n := n) D s f σ η) ≤ |
| T*averagedCorrelation (lowGradient (m := m) (n := n) D s f σ η)+ |
| a*averagedEnergy (lowGradient (m := m) (n := n) D s f σ η)+ |
| (σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)*thirdError (m := m) (n := n) f σ η := by |
| have hi (B : FaceArray A m n) : |
| Integrable (fun c => 𝔼 x : Point m n, |
| lowGradient D s f σ η B c x*gradient f B σ η c x) (gauss (Point m n)) := by |
| simp only [Fintype.expect_eq_sum_div_card] |
| exact (integrable_finsetSum _ (fun x _ => |
| (lowGradient_memLp D s f σ η B x).integrable_mul (gradient_memLp f B hσ η x))).div_const _ |
| have he (B : FaceArray A m n) : |
| Integrable (fun c => spatialEnergy (lowGradient D s f σ η B c)) (gauss (Point m n)) := by |
| simp only [spatialEnergy] |
| exact (integrable_finsetSum _ (fun x _ => (lowGradient_memLp D s f σ η B x).integrable_sq)).const_mul _ |
| have hp (B : FaceArray A m n) := integral_mono (hi B) |
| (((lowGradient_correlation_integrable D s f σ η B).const_mul T).add ((he B).const_mul a) |>.add |
| ((thirdRejection_integrable f B σ η).const_mul ((σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)))) |
| (fun c => gradient_correlation_truncate f B hn hσ hη hT ha c (lowGradient D s f σ η B c)) |
| have hp' (B : FaceArray A m n) : |
| (∫ c, (𝔼 x : Point m n, lowGradient D s f σ η B c x*gradient f B σ η c x) |
| ∂gauss (Point m n)) ≤ |
| T*(∫ c, faceCorrelation (lowGradient D s f σ η B c) ∂gauss (Point m n))+ |
| a*(∫ c, spatialEnergy (lowGradient D s f σ η B c) ∂gauss (Point m n))+ |
| (σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)* |
| (∫ c, thirdRejection f B σ η c ∂gauss (Point m n)) := by |
| have hh := hp B |
| change _ ≤ ∫ c, (T*faceCorrelation (lowGradient D s f σ η B c)+ |
| a*spatialEnergy (lowGradient D s f σ η B c)+ |
| (σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)*thirdRejection f B σ η c) |
| ∂gauss (Point m n) at hh |
| rw [integral_add (f := fun c => T*faceCorrelation (lowGradient D s f σ η B c)+a*spatialEnergy (lowGradient D s f σ η B c)) |
| (g := fun c => (σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)*thirdRejection f B σ η c) |
| ((lowGradient_correlation_integrable D s f σ η B).const_mul T |>.add ((he B).const_mul a)) |
| ((thirdRejection_integrable f B σ η).const_mul _), |
| integral_add ((lowGradient_correlation_integrable D s f σ η B).const_mul T) ((he B).const_mul a)] at hh |
| simpa only [integral_const_mul] using hh |
| have hh := Finset.expect_le_expect (s := Finset.univ) (fun B _ => hp' B) |
| rw [lowGradient_inner_identity D s f hσ η] at hh |
| simpa only [Finset.expect_add_distrib,← Finset.mul_expect,averagedCorrelation,averagedEnergy,thirdError] |
| using hh |
| end MinUncut.Inner |
|
|
| namespace MinUncut.Inner |
| open MeasureTheory ProbabilityTheory GaussianHermite RowNoise |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {Ξ : Type*} [Fintype Ξ] (V A : Ξ → Type*) |
| [∀ ξ, AddCommGroup (V ξ)] [∀ ξ, Module F₂ (V ξ)] [∀ ξ, AddTorsor (V ξ) (A ξ)] |
| [∀ ξ, Fintype (A ξ)] {m n : ℕ} |
|
|
| lemma lowGradient_correlation_truncated_weighted (D s : ℕ) |
| (w : Ξ → ℝ) (hw : ∀ ξ,0≤w ξ) (f : ∀ ξ, FoldedProof (A ξ)) |
| (hn : 0<n) {σ η T a : ℝ} (hσ : σ≠0) (hη : 0<η) (hT : 0<T) (ha : 0<a) : |
| (∑ ξ, w ξ*averagedEnergy (lowGradient (m := m) (n := n) D s (f ξ) σ η)) ≤ |
| T*(∑ ξ, w ξ*averagedCorrelation (lowGradient (m := m) (n := n) D s (f ξ) σ η))+ |
| a*(∑ ξ, w ξ*averagedEnergy (lowGradient (m := m) (n := n) D s (f ξ) σ η))+ |
| (σ⁻¹^2/(4*a*T))*((Fintype.card (Code m n):ℝ)/η)* |
| (∑ ξ, w ξ*thirdError (m := m) (n := n) (f ξ) σ η) := by |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun ξ _ => mul_le_mul_of_nonneg_left |
| (lowGradient_correlation_truncated (m := m) D s (f ξ) hn hσ hη hT ha) (hw ξ)) |
| convert hh using 1 |
| simp only [mul_add,Finset.sum_add_distrib] |
| congr 2 <;> rw [Finset.mul_sum] <;> apply Finset.sum_congr rfl <;> intro ξ _ <;> ring |
| end MinUncut.Inner |
|
|
| end |
| end |
|
|
| end OAI |
|
|
