| import Mathlib |
| import OAI.Computability.MinUncut.Analysis.GradientEnergyFirstTables |
| import OAI.Computability.MinUncut.Analysis.LowGradient |
|
|
| 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 GaussianHermite RowNoise |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
| {m n : ℕ} |
|
|
| def firstError (f : FoldedProof A) (σ η : ℝ) : ℝ := |
| 𝔼 B : FaceArray A m n, ∫ c, firstRejection f B σ η c ∂gauss (Point m n) |
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| def secondError (f : FoldedProof A) (a σ η : ℝ) : ℝ := |
| 𝔼 B : FaceArray A m n, 𝔼 C : FaceArray A m n, |
| RowNoise.density a B C*(∫ c, secondRejection f B C σ η c ∂gauss (Point m n)) |
|
|
| lemma gaussian_tail_of_cutoff (f : FoldedProof A) (hn : 0 < n) |
| {σ ε : ℝ} (hσ : σ≠0) (hε : 0≤ε) (η : ℝ) (s : ℕ) |
| (hs : ∀ k : ℕ, s+2≤k → (k:ℝ)*(smoothingRho σ)^(2*k)≤ε) : |
| averagedEnergy (m := m) (n := n) |
| (fun B c x => gradient f B σ η c x-gaussianProject s f B σ η c x) ≤ ε := by |
| have he (B : FaceArray A m n) : |
| (∫ c, spatialEnergy (fun x => gradient f B σ η c x-gaussianProject s f B σ η c x) |
| ∂gauss (Point m n)) ≤ ε := by |
| rw [gaussian_residual_energy s f B hn hσ] |
| exact convolution_gradient_tail (ne_of_gt (smoothingRho_pos σ)) hσ |
| (smoothingRho_rotation σ) hε s hs (measurable_tableSmooth f B η) |
| (tableSmooth_abs_le f B η) |
| simpa [averagedEnergy] using Finset.expect_le_expect (s := Finset.univ) (fun B _ => he B) |
| |
| theorem lowGradient_energy_lower (f : FoldedProof A) (hn : 0 < n) |
| {σ η a ε : ℝ} (hσ : σ≠0) (hη : η≠0) (ha : 0≤a) (ha1 : a≤1) (hε : 0≤ε) |
| (D s : ℕ) (hD : ∀ k : ℕ, D<k → 1≤(k:ℝ)*a) |
| (hs : ∀ k : ℕ, s+2≤k → (k:ℝ)*(smoothingRho σ)^(2*k)≤ε) : |
| 1-4*firstError (m := m) (n := n) f σ η- |
| 8*σ⁻¹^2*secondError (m := m) (n := n) f a σ η-2*ε ≤ |
| averagedEnergy (lowGradient (m := m) (n := n) D s f σ η) := by |
| have h1 := gradient_energy_first_tables (m := m) f hn hσ hη |
| have h2 := gradient_row_tail_average (m := m) f hn hσ ha ha1 η D hD |
| have hc := gaussian_tail_of_cutoff (m := m) f hn hσ hε η s hs |
| have he := lowGradient_residual_le (m := m) (n := n) D s f hσ η |
| have hi := lowGradient_energy_identity (m := m) (n := n) D s f hσ η |
| change 1-4*firstError (m := m) (n := n) f σ η ≤ |
| averagedEnergy (m := m) (n := n) (fun B c => gradient f B σ η c) at h1 |
| change averagedEnergy (m := m) (n := n) (fun B c x => |
| gradient f B σ η c x-rowProjectedGradient D f σ η B c x) ≤ |
| 4*σ⁻¹^2*secondError (m := m) (n := n) f a σ η at h2 |
| linarith |
| end MinUncut.Inner |
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|
| end |
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| end OAI |
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