| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Convolution |
| import OAI.Computability.MinUncut.Estimates.CoupledAnswer |
|
|
| namespace OAI |
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|
| 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 |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
| variable {m n : ℕ} |
|
|
| def rowProjectedGradient (D : ℕ) (f : FoldedProof A) (σ η : ℝ) |
| (B : FaceArray A m n) (c : Point m n → ℝ) (x : Point m n) : ℝ := |
| RowNoise.project D (fun C => gradient f C σ η c x) B |
| |
| theorem gradient_row_tail (f : FoldedProof A) (hn : 0 < n) |
| {σ a : ℝ} (hσ : σ≠0) (ha : 0≤a) (ha1 : a≤1) (η : ℝ) (D : ℕ) |
| (hD : ∀ k : ℕ, D < k → 1≤(k:ℝ)*a) (c : Point m n → ℝ) : |
| (𝔼 B : FaceArray A m n, spatialEnergy (fun x => |
| gradient f B σ η c x-rowProjectedGradient D f σ η B c x)) ≤ |
| 4*σ⁻¹^2*(𝔼 B : FaceArray A m n, 𝔼 C : FaceArray A m n, |
| RowNoise.density a B C*secondRejection f B C σ η c) := by |
| have htail (x : Point m n) := RowNoise.row_tail_le ha ha1 D hD |
| (fun B : FaceArray A m n => gradient f B σ η c x) |
| calc |
| _ = ((n^m:ℕ):ℝ)⁻¹*∑ x : Point m n, |
| 𝔼 B : FaceArray A m n, (gradient f B σ η c x-rowProjectedGradient D f σ η B c x)^2 := by |
| simp only [spatialEnergy,← Finset.mul_expect,Finset.expect_sum_comm] |
| _ ≤ ((n^m:ℕ):ℝ)⁻¹*∑ x : Point m n, |
| 𝔼 B : FaceArray A m n, 𝔼 C : FaceArray A m n, |
| RowNoise.density a B C*(gradient f B σ η c x-gradient f C σ η c x)^2 := |
| mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun x _ => htail x)) (by positivity) |
| _ = 𝔼 B : FaceArray A m n, 𝔼 C : FaceArray A m n, |
| RowNoise.density a B C*spatialEnergy |
| (fun x => gradient f B σ η c x-gradient f C σ η c x) := by |
| simp only [spatialEnergy,Finset.mul_sum,Finset.expect_sum_comm, |
| ← Finset.mul_expect,mul_left_comm] |
| _ ≤ 𝔼 B : FaceArray A m n, 𝔼 C : FaceArray A m n, |
| RowNoise.density a B C*(4*σ⁻¹^2*secondRejection f B C σ η c) := |
| Finset.expect_le_expect (fun B _ => Finset.expect_le_expect (fun C _ => |
| mul_le_mul_of_nonneg_left (gradient_second_comparison f B C hn hσ η c) |
| (RowNoise.density_nonneg ha ha1 B C))) |
| _ = _ := by |
| have he (B C : FaceArray A m n) : |
| RowNoise.density a B C*(4*σ⁻¹^2*secondRejection f B C σ η c) = |
| (4*σ⁻¹^2)*(RowNoise.density a B C*secondRejection f B C σ η c) := by ring |
| simp_rw [he,← Finset.mul_expect] |
| end MinUncut.Inner |
|
|
| end |
|
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| end OAI |
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