| import OAI.MeasureTheory.Falconer.Estimates.RegularizationPairBounds | |
| import OAI.MeasureTheory.Falconer.Shells.OriginalClassDistance | |
| import OAI.MeasureTheory.Falconer.Estimates.ClassPairWeights | |
| namespace OAI | |
| /-! # Exact original-mass mixture for the two-class distance law -/ | |
| namespace Falconer.Regularization | |
| open Set MeasureTheory | |
| open scoped ENNReal NNReal | |
| variable {d : ℕ} | |
| noncomputable def classPairWeight (Q₁ Q₂ : DyadicCube d) (μ ν : Measure (Space d)) | |
| (N B₁ B₂ : ℕ) (i : List ℕ × List ℕ) : ℝ≥0 := | |
| (classMeasure Q₁ μ (regularizationLeaves Q₁ μ N) (regularizationLabel Q₁ μ N B₁) i.1 univ * | |
| classMeasure Q₂ ν (regularizationLeaves Q₂ ν N) (regularizationLabel Q₂ ν N B₂) i.2 univ).toNNReal | |
| theorem classPairWeight_sum_le_one (Q₁ Q₂ : DyadicCube d) | |
| (hQ₁ : 0 < Q₁.side) (hQ₂ : 0 < Q₂.side) | |
| (μ ν : Measure (Space d)) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| (N B₁ B₂ : ℕ) (t₁ t₂ : ℝ≥0∞) : | |
| (∑ i ∈ (regularizationIndex Q₁ μ N B₁ t₁).product (regularizationIndex Q₂ ν N B₂ t₂), | |
| (classPairWeight Q₁ Q₂ μ ν N B₁ B₂ i : ℝ)) ≤ 1 := | |
| retained_class_pair_real_mass_sum_le_one Q₁ Q₂ hQ₁ hQ₂ μ ν | |
| (Finset.filter_subset _ _) (Finset.filter_subset _ _) _ _ _ _ | |
| (Finset.filter_subset _ _) (Finset.filter_subset _ _) | |
| theorem classPairWeight_coe (Q₁ Q₂ : DyadicCube d) (μ ν : Measure (Space d)) | |
| [IsFiniteMeasure μ] [IsFiniteMeasure ν] (N B₁ B₂ : ℕ) (i : List ℕ × List ℕ) : | |
| (classPairWeight Q₁ Q₂ μ ν N B₁ B₂ i : ℝ≥0∞) = | |
| classMeasure Q₁ μ (regularizationLeaves Q₁ μ N) (regularizationLabel Q₁ μ N B₁) i.1 univ * | |
| classMeasure Q₂ ν (regularizationLeaves Q₂ ν N) (regularizationLabel Q₂ ν N B₂) i.2 univ := by | |
| apply ENNReal.coe_toNNReal | |
| apply ENNReal.mul_ne_top | |
| · change μ.restrict _ univ ≠ ∞ | |
| exact measure_ne_top _ _ | |
| · change ν.restrict _ univ ≠ ∞ | |
| exact measure_ne_top _ _ | |
| noncomputable def regularizationPairDistance (Q₁ Q₂ : DyadicCube d) (μ ν : Measure (Space d)) | |
| (N B₁ B₂ : ℕ) (t₁ t₂ : ℝ≥0∞) (σ κ : ℝ) | |
| (p₁ p₂ : List ℕ → CellPath d) (c : List ℕ × List ℕ → ℕ) | |
| (Γ : Set (Space d × Space d)) : Measure ℝ := | |
| partitionEndpointDistance μ ν Γ | |
| ((regularizationIndex Q₁ μ N B₁ t₁).product (regularizationIndex Q₂ ν N B₂ t₂)) | |
| (fun i => regularizationSet Q₁ μ N B₁ i.1) (fun i => regularizationSet Q₂ ν N B₂ i.2) | |
| (fun i => sourceOuterMask Q₁ μ N B₁ σ κ (p₁ i.1) (c i)) | |
| (fun i => sourceOuterMask Q₂ ν N B₂ σ κ (p₂ i.2) (c i)) | |
| theorem regularizationPairDistance_eq_mixture {k : ℕ} | |
| (Q₁ Q₂ : DyadicCube d) (μ ν : Measure (Space d)) | |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| (N B₁ B₂ : ℕ) (t₁ t₂ : ℝ≥0∞) (ht₁ : 0 < t₁) (ht₂ : 0 < t₂) (σ κ : ℝ) | |
| (p₁ p₂ : List ℕ → CellPath d) | |
| (he₁ : ∀ T ∈ regularizationIndex Q₁ μ N B₁ t₁, regularizationLabel Q₁ μ N B₁ (p₁ T) = T) | |
| (he₂ : ∀ T ∈ regularizationIndex Q₂ ν N B₂ t₂, regularizationLabel Q₂ ν N B₂ (p₂ T) = T) | |
| (c : List ℕ × List ℕ → ℕ) (A D : Fin k → Set (Space d)) | |
| (hm : ∀ j, MeasurableSet (A j) ∧ MeasurableSet (D j)) | |
| (hd : Pairwise fun j l => Disjoint (A j ×ˢ D j) (A l ×ˢ D l)) : | |
| regularizationPairDistance Q₁ Q₂ μ ν N B₁ B₂ t₁ t₂ σ κ p₁ p₂ c (⋃ j, A j ×ˢ D j) = | |
| ∑ i ∈ (regularizationIndex Q₁ μ N B₁ t₁).product (regularizationIndex Q₂ ν N B₂ t₂), | |
| classPairWeight Q₁ Q₂ μ ν N B₁ B₂ i • | |
| ∑ j, endpointWeightedDistance ((sourceClass Q₁ μ N B₁ (p₁ i.1)).restrict (A j)) | |
| ((sourceClass Q₂ ν N B₂ (p₂ i.2)).restrict (D j)) 1 1 1 | |
| (sourceOuterMask Q₁ μ N B₁ σ κ (p₁ i.1) (c i)) | |
| (sourceOuterMask Q₂ ν N B₂ σ κ (p₂ i.2) (c i)) := by | |
| unfold regularizationPairDistance | |
| rw [partitionEndpointDistance_product_sum μ ν A D hm hd] | |
| · apply Finset.sum_congr rfl | |
| intro i hi | |
| have hi₁ := (Finset.mem_product.mp hi).1 | |
| have hi₂ := (Finset.mem_product.mp hi).2 | |
| rw [Finset.smul_sum] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| simp only [regularizationSet] | |
| rw [original_class_endpoint_distance Q₁ Q₂ μ ν _ _ _ _ i.1 i.2 | |
| (ht₁.trans_le (Finset.mem_filter.mp hi₁).2).ne' | |
| (ht₂.trans_le (Finset.mem_filter.mp hi₂).2).ne' (A j) (D j) (hm j).1 (hm j).2] | |
| rw [← sourceClass_eq_normalized Q₁ μ N B₁ (p₁ i.1) i.1 (he₁ i.1 hi₁), | |
| ← sourceClass_eq_normalized Q₂ ν N B₂ (p₂ i.2) i.2 (he₂ i.2 hi₂)] | |
| rw [← classPairWeight_coe Q₁ Q₂ μ ν N B₁ B₂ i] | |
| rfl | |
| · exact fun _ _ => regularizationSet_measurable _ _ _ _ _ | |
| · exact fun _ _ => regularizationSet_measurable _ _ _ _ _ | |
| · exact fun _ _ => regularOuterMask_measurable _ _ _ _ _ _ _ | |
| · exact fun _ _ => regularOuterMask_measurable _ _ _ _ _ _ _ | |
| · exact fun _ _ _ _ => (regularOuterMask_bounds _ _ _ _ _ _ _ _ _).1 | |
| · exact fun _ _ _ _ => (regularOuterMask_bounds _ _ _ _ _ _ _ _ _).1 | |
| end Falconer.Regularization | |
| end OAI | |