| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.F2 |
| import OAI.Computability.MinUncut.Analysis.Gradient |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Polynomial Filter |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory WithLp |
| open scoped BigOperators RealInnerProductSpace |
| open scoped BigOperators |
| namespace MinUncut.Inner |
| open MeasureTheory ProbabilityTheory GaussianHermite |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| attribute [local irreducible] query pullQuery |
| variable {m n : ℕ} |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
|
|
| def coupledAnswer (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) (p : (Point m n → ℝ) × (Code m n → ℝ)) : ℝ := |
| bitSign (f.answer (pullQuery B (c+σ•p.1) (η•p.2))) |
|
|
| def secondFailure (f : FoldedProof A) (B C : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) (p : (Point m n → ℝ) × (Code m n → ℝ)) : ℝ := |
| if f.answer (pullQuery B (c+σ•p.1) (η•p.2)) = |
| f.answer (pullQuery C (c+σ•p.1) (η•p.2)) then 0 else 1 |
|
|
| def secondRejection (f : FoldedProof A) (B C : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) : ℝ := |
| ∫ p, secondFailure f B C σ η c p ∂(gauss (Point m n)).prod (gauss (Code m n)) |
|
|
| lemma coupledAnswer_measurable (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) : Measurable (coupledAnswer f B σ η c) := by |
| have hg : Measurable (fun p : (Point m n → ℝ) × (Code m n → ℝ) => (c+σ•p.1,p.2)) := by fun_prop |
| exact (measurable_answer f B η).comp hg |
|
|
| lemma coupledAnswer_memLp (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) : |
| MemLp (coupledAnswer f B σ η c) 2 ((gauss (Point m n)).prod (gauss (Code m n))) := |
| MemLp.of_bound (coupledAnswer_measurable f B σ η c).aestronglyMeasurable 1 |
| (ae_of_all _ (fun p => by simp only [Real.norm_eq_abs,coupledAnswer,bitSign_abs,le_refl])) |
|
|
| omit [Fintype A] in |
| lemma secondFailure_eq (f : FoldedProof A) (B C : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) (p : (Point m n → ℝ) × (Code m n → ℝ)) : |
| (coupledAnswer f B σ η c p-coupledAnswer f C σ η c p)^2 = 4*secondFailure f B C σ η c p := by |
| unfold coupledAnswer secondFailure |
| cases f.answer (pullQuery B (c+σ•p.1) (η•p.2)) <;> |
| cases f.answer (pullQuery C (c+σ•p.1) (η•p.2)) <;> norm_num [bitSign] |
|
|
| lemma coupled_score_integrable (f : FoldedProof A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) (x : Point m n) : |
| Integrable (fun p => p.1 x*coupledAnswer f B σ η c p) |
| ((gauss (Point m n)).prod (gauss (Code m n))) := |
| ((GaussianSmoothing.coordinate_memLp_two x).comp_fst (gauss (Code m n))).integrable_mul |
| (coupledAnswer_memLp f B σ η c) |
|
|
| lemma gradient_coupled_score (f : FoldedProof A) (B : FaceArray A m n) |
| {σ : ℝ} (hσ : σ≠0) (η : ℝ) (c : Point m n → ℝ) (x : Point m n) : |
| gradient f B σ η c x=Real.sqrt (n^m:ℕ)/σ* |
| ∫ p, p.1 x*coupledAnswer f B σ η c p ∂(gauss (Point m n)).prod (gauss (Code m n)) := by |
| rw [integral_prod _ (coupled_score_integrable f B σ η c x)] |
| exact gradient_convolution f B hσ η c x |
|
|
| theorem gradient_second_comparison (f : FoldedProof A) (B C : FaceArray A m n) |
| (hn : 0 < n) {σ : ℝ} (hσ : σ≠0) (η : ℝ) (c : Point m n → ℝ) : |
| spatialEnergy (fun x => gradient f B σ η c x-gradient f C σ η c x) ≤ |
| 4*σ⁻¹^2*secondRejection f B C σ η c := by |
| let μ := (gauss (Point m n)).prod (gauss (Code m n)) |
| let D := fun p => coupledAnswer f B σ η c p-coupledAnswer f C σ η c p |
| have hD : MemLp D 2 μ := (coupledAnswer_memLp f B σ η c).sub (coupledAnswer_memLp f C σ η c) |
| have hX (x : Point m n) : MemLp (fun p => p.1 x) 2 μ := |
| (GaussianSmoothing.coordinate_memLp_two x).comp_fst (gauss (Code m n)) |
| have hXX (x y : Point m n) : (∫ p, p.1 x*p.1 y ∂μ)=if x=y then 1 else 0 := by |
| have hi : Integrable (fun p : (Point m n → ℝ) × (Code m n → ℝ) => p.1 x*p.1 y) μ := |
| (hX x).integrable_mul (hX y) |
| change (∫ p, p.1 x*p.1 y ∂(gauss (Point m n)).prod (gauss (Code m n)))=_ |
| rw [integral_prod _ hi] |
| simpa only [integral_const,MeasureTheory.probReal_univ,one_smul,gauss,GaussianSmoothing.gamma] using GaussianSmoothing.coordinate_second_moment x y |
| have hS := GaussianSmoothing.integral_bessel hX hXX hD |
| have hN : (0 : ℝ)<(n^m:ℕ) := by exact_mod_cast pow_pos hn m |
| have hd (x : Point m n) : |
| gradient f B σ η c x-gradient f C σ η c x = |
| Real.sqrt (n^m:ℕ)/σ*(∫ p, p.1 x*D p ∂μ) := by |
| rw [gradient_coupled_score f B hσ,gradient_coupled_score f C hσ] |
| simp only [D,mul_sub] |
| rw [integral_sub (coupled_score_integrable f B σ η c x) (coupled_score_integrable f C σ η c x)] |
| ring |
| have he : spatialEnergy (fun x => gradient f B σ η c x-gradient f C σ η c x) = |
| σ⁻¹^2*∑ x, (∫ p, p.1 x*D p ∂μ)^2 := by |
| simp only [spatialEnergy,hd,mul_pow,div_pow,Real.sq_sqrt hN.le,← Finset.mul_sum] |
| field_simp |
| rw [he] |
| calc |
| _ ≤ σ⁻¹^2*(∫ p, (D p)^2 ∂μ) := mul_le_mul_of_nonneg_left hS (sq_nonneg _) |
| _ = _ := by |
| simp_rw [D,secondFailure_eq] |
| rw [integral_const_mul] |
| change σ⁻¹^2*(4*secondRejection f B C σ η c)=_ |
| ring |
| end MinUncut.Inner |
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|
| end |
|
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| end OAI |
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