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7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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
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