| import OAI.Geometry.RCD.Transport.AsymmetricMidpointReplacement |
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| namespace OAI |
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| /-! # An actual compressed midpoint with unequal endpoint bounds |
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| Minimizing excess mass and replacing its saturated region turns the CD |
| support estimate into a density bound. Separate endpoint bounds allow |
| one marginal to concentrate while retaining the contribution of the other. |
| -/ |
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| open Set Filter Metric MeasureTheory TopologicalSpace |
| open scoped ENNReal |
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| namespace RegularPointCoordinates |
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| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| attribute [local instance] pathBorel pathBorelSpace |
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| theorem FullCD.improve_localized_saturation_asymmetric |
| (μ : Measure X) [SigmaFinite μ] |
| (W : Set X) (hW : MeasurableSet W) [IsFiniteMeasure (μ.restrict W)] |
| (K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X) |
| [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁] |
| {C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0) |
| (h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ) |
| (hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁) |
| (s₀ s₁ : Set X) (hs0 : ∀ᵐ x ∂ν₀, x ∈ s₀) (hs1 : ∀ᵐ x ∂ν₁, x ∈ s₁) |
| (hn : 2 ≤ n) (D : ℝ) |
| (hD : ∀ x ∈ s₀, ∀ y ∈ s₁, dist x y ≤ D) |
| (hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → |
| (∀ᵐ γ ∂q, IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → |
| ∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W) |
| (hfill : C⁻¹ ≤ midpointSupportFactor K n D C₀ C₁) |
| (π : Measure C(Time, X)) [IsProbabilityMeasure π] |
| (hg : ∀ᵐ γ ∂π, IsGeodesic γ) |
| (hp : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1)))) |
| (hpos : transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≠ 0) |
| (hmin : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' → |
| (∀ᵐ γ ∂π', IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) → |
| transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≤ |
| transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C) : |
| ∃ π' : Measure C(Time, X), IsProbabilityMeasure π' ∧ |
| (∀ᵐ γ ∂π', IsGeodesic γ) ∧ |
| IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) ∧ |
| transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C = |
| transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ∧ |
| (μ.restrict W) {x | C < (π.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} + |
| transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C / 2 * C⁻¹ < |
| (μ.restrict W) {x | C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} := by |
| let ν := π.map (fun γ => γ midpointTime) |
| have hvprob : IsProbabilityMeasure ν := |
| inferInstance |
| let e := transportExcessMass (μ.restrict W) ν C |
| let v := (μ.restrict W) {x | C < ν.rnDeriv (μ.restrict W) x} |
| have hetop : e ≠ ∞ := ne_top_of_le_ne_top (measure_ne_top ν univ) |
| (transportExcessMass_le (μ.restrict W) ν C) |
| obtain ⟨s, hs, hμs, hνs⟩ := (Measure.mutuallySingular_singularPart ν (μ.restrict W)).symm |
| have hthreshold : e / 2 + C * v < ν (s ∪ {x | C < ν.rnDeriv (μ.restrict W) x}) := by |
| rw [positive_region_mass (μ.restrict W) ν hC hs hμs hνs] |
| exact ENNReal.add_lt_add_right |
| (ENNReal.mul_ne_top hC (measure_ne_top (μ.restrict W) _)) (ENNReal.half_lt_self hpos hetop) |
| obtain ⟨δ, hδ0, hδtop, hδmass⟩ := |
| exists_buffer_positive_region (μ.restrict W) ν hC s hthreshold |
| let B := s ∪ {x | C + δ < ν.rnDeriv (μ.restrict W) x} |
| have hB : MeasurableSet B := hs.union |
| (measurableSet_lt measurable_const (ν.measurable_rnDeriv (μ.restrict W))) |
| have hB0 : ν B ≠ 0 := (lt_of_le_of_lt zero_le hδmass).ne' |
| let a := δ / (C + δ) |
| have ha : a ≤ 1 := ENNReal.div_le_of_le_mul |
| (by simp) |
| have ha0 : 0 < a := ENNReal.div_pos hδ0.ne' (ENNReal.add_ne_top.mpr ⟨hC, hδtop⟩) |
| have hrem : a • ν.restrict B ≤ ν - C • μ.restrict W := |
| buffer_fraction_le_excess (μ.restrict W) ν hC hδ0 hδtop hs hμs |
| let : IsFiniteMeasure (a • ν.restrict B) := |
| isFiniteMeasure_of_le ν (hrem.trans Measure.sub_le) |
| obtain ⟨π', η, hπ', hη, hg', hp', heq, hmass, hvol⟩ := |
| hcd.replace_midpoint_region_asymmetric μ K n ν₀ ν₁ hC₀ hC₁ h0 h1 hm0 hm1 s₀ s₁ hs0 hs1 |
| hn D hD π hg hp B hB hB0 ha0 ha |
| let := hπ' |
| let := hη |
| have hηle : η ≤ π'.map (fun γ => γ midpointTime) := by |
| rw [heq] |
| exact Measure.le_add_left le_rfl |
| have hηsupp : ∀ᵐ x ∂η, x ∈ W := |
| (hsupport π' hπ' hg' hp').filter_mono hηle.absolutelyContinuous.ae_le |
| have hvolLoc := hvol.trans_eq (ac_support_volume_restrict μ η hW hηsupp) |
| have hmin' := hmin π' hπ' hg' hp' |
| have hmass' : η univ = (a • ν.restrict B) univ := by |
| simpa only [Measure.smul_apply, smul_eq_mul, Measure.restrict_apply_univ] using hmass |
| have hequal : transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C = e := by |
| apply le_antisymm _ hmin' |
| rw [heq] |
| exact (transportExcessMass_add_le (μ.restrict W) (ν - a • ν.restrict B) η hC).trans_eq (by |
| rw [hmass', transportExcessMass_remove (μ.restrict W) ν (a • ν.restrict B) hC hrem]) |
| have hsaturated : ∀ᵐ x ∂μ.restrict W, η.rnDeriv (μ.restrict W) x ≠ 0 → |
| C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x := by |
| rw [heq] |
| apply ac_support_saturated_of_minimizing_replacement (μ.restrict W) ν (a • ν.restrict B) η |
| hC hrem hmass' |
| simpa only [heq] using hmin' |
| refine ⟨π', hπ', hg', hp', hequal, ?_⟩ |
| have hgain : v + e / 2 * C⁻¹ < ν B * C⁻¹ := by |
| have hh := ENNReal.mul_lt_mul_left (ENNReal.inv_ne_zero.mpr hC) |
| (ENNReal.inv_ne_top.mpr hCpos) hδmass |
| convert hh using 1 |
| rw [add_mul, mul_assoc, mul_comm v, ← mul_assoc, ENNReal.mul_inv_cancel hCpos hC] |
| simp only [one_mul, add_comm] |
| exact hgain.trans_le (calc |
| ν B * C⁻¹ ≤ ν B * (midpointSupportFactor K n D C₀ C₁) := |
| mul_le_mul_right hfill _ |
| _ = midpointSupportFactor K n D C₀ C₁ * ν B := by ac_rfl |
| _ ≤ (μ.restrict W) {x | η.rnDeriv (μ.restrict W) x ≠ 0} := hvolLoc |
| _ ≤ (μ.restrict W) {x | C < (π'.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} := |
| measure_mono_ae hsaturated) |
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| theorem FullCD.localized_excess_minimizer_zero_asymmetric |
| (μ : Measure X) [SigmaFinite μ] |
| (W : Set X) (hW : MeasurableSet W) [IsFiniteMeasure (μ.restrict W)] |
| (K : ℝ) (n : ℕ) (hcd : FullCD μ K n) (ν₀ ν₁ : Measure X) |
| [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁] |
| {C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0) |
| (h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ) |
| (hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁) |
| (s₀ s₁ : Set X) (hs0 : ∀ᵐ x ∂ν₀, x ∈ s₀) (hs1 : ∀ᵐ x ∂ν₁, x ∈ s₁) |
| (hn : 2 ≤ n) (D : ℝ) |
| (hD : ∀ x ∈ s₀, ∀ y ∈ s₁, dist x y ≤ D) |
| (hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → |
| (∀ᵐ γ ∂q, IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → |
| ∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W) |
| (hfill : C⁻¹ ≤ midpointSupportFactor K n D C₀ C₁) |
| (π : Measure C(Time, X)) [IsProbabilityMeasure π] |
| (hg : ∀ᵐ γ ∂π, IsGeodesic γ) |
| (hp : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1)))) |
| (hmin : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' → |
| (∀ᵐ γ ∂π', IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) → |
| transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C ≤ |
| transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C) : |
| transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C = 0 := by |
| by_contra hpos |
| let e := transportExcessMass (μ.restrict W) (π.map (fun γ => γ midpointTime)) C |
| let A := {ρ : Measure C(Time, X) |
| (∀ᵐ γ ∂ρ, IsGeodesic γ) ∧ |
| IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0, γ 1))) ∧ |
| transportExcessMass (μ.restrict W) (ρ.map (fun γ => γ midpointTime)) C = e} |
| have : Nonempty A := ⟨⟨π, inferInstance, hg, hp, rfl⟩⟩ |
| let v (ρ : A) := (μ.restrict W) {x | C < (ρ.val.map (fun γ => γ midpointTime)).rnDeriv (μ.restrict W) x} |
| let S := ⨆ ρ : A, v ρ |
| have hS : S ≠ ∞ := ne_top_of_le_ne_top (measure_ne_top (μ.restrict W) univ) |
| (iSup_le fun _ => measure_mono (subset_univ _)) |
| have hstep : S + e / 2 * C⁻¹ ≤ S := by |
| rw [ENNReal.iSup_add] |
| apply iSup_le |
| intro ρ |
| let : IsProbabilityMeasure ρ.val := ρ.property.1 |
| have hposρ : transportExcessMass (μ.restrict W) (ρ.val.map (fun γ => γ midpointTime)) C ≠ 0 := by |
| rw [ρ.property.2.2.2] |
| exact hpos |
| have hminρ : ∀ (π' : Measure C(Time, X)), IsProbabilityMeasure π' → |
| (∀ᵐ γ ∂π', IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (π'.map (fun γ => (γ 0, γ 1))) → |
| transportExcessMass (μ.restrict W) (ρ.val.map (fun γ => γ midpointTime)) C ≤ |
| transportExcessMass (μ.restrict W) (π'.map (fun γ => γ midpointTime)) C := by |
| intro π' hπ' hg' hp' |
| rw [ρ.property.2.2.2] |
| exact hmin π' hπ' hg' hp' |
| obtain ⟨π', hπ', hg', hp', hequal, hgain⟩ := |
| hcd.improve_localized_saturation_asymmetric μ W hW K n ν₀ ν₁ |
| hC₀ hC₁ hC hCpos h0 h1 hm0 hm1 s₀ s₁ hs0 hs1 hn D hD hsupport hfill |
| ρ.val ρ.property.2.1 ρ.property.2.2.1 hposρ hminρ |
| let ρ' : A := ⟨π', hπ', hg', hp', hequal.trans ρ.property.2.2.2⟩ |
| rw [ρ.property.2.2.2] at hgain |
| exact hgain.le.trans (le_iSup v ρ') |
| have hδ : 0 < e / 2 * C⁻¹ := ENNReal.mul_pos |
| (ENNReal.div_pos hpos (by simp)).ne' (ENNReal.inv_ne_zero.mpr hC) |
| have hstrict := ENNReal.add_lt_add_left hS hδ |
| rw [add_zero] at hstrict |
| exact (not_lt_of_ge hstep) hstrict |
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|
| section Attainment |
| variable [ProperSpace X] [SeparableSpace X] |
|
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| theorem FullCD.exists_compressed_midpoint_asymmetric |
| (μ : Measure X) [SigmaFinite μ] (K : ℝ) (n : ℕ) (hcd : FullCD μ K n) |
| (ν₀ ν₁ : Measure X) [IsProbabilityMeasure ν₀] [IsProbabilityMeasure ν₁] |
| {C₀ C₁ C : ℝ≥0∞} (hC₀ : C₀ ≠ ∞) (hC₁ : C₁ ≠ ∞) (hC : C ≠ ∞) (hCpos : C ≠ 0) |
| (h0 : ν₀ ≤ C₀ • μ) (h1 : ν₁ ≤ C₁ • μ) |
| (hm0 : FiniteSecondMoment ν₀) (hm1 : FiniteSecondMoment ν₁) |
| (p : X) {R : ℝ} (hR : 0 ≤ R) |
| [IsFiniteMeasure (μ.restrict (closedBall p (3 * R)))] |
| [Measure.OuterRegular (μ.restrict (closedBall p (3 * R)))] |
| (hs0 : ∀ᵐ x ∂ν₀, dist x p ≤ R) (hs1 : ∀ᵐ x ∂ν₁, dist x p ≤ R) |
| (hn : 2 ≤ n) |
| (hfill : C⁻¹ ≤ midpointSupportFactor K n (2 * R) C₀ C₁) : |
| ∃ π : Measure C(Time, X), IsProbabilityMeasure π ∧ |
| (∀ᵐ γ ∂π, IsGeodesic γ) ∧ |
| IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0, γ 1))) ∧ |
| π.map (fun γ => γ midpointTime) ≤ C • μ := by |
| let W := closedBall p (3 * R) |
| have hW : MeasurableSet W := isClosed_closedBall.measurableSet |
| have hsupport : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → |
| (∀ᵐ γ ∂q, IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → |
| ∀ᵐ x ∂q.map (fun γ => γ midpointTime), x ∈ W := by |
| intro q hq hgq hpq |
| apply (ae_map_iff (measurable_path_eval midpointTime).aemeasurable hW).mpr |
| exact (optimal_geodesic_plan_bounded ν₀ ν₁ q hgq hpq p hR hs0 hs1).mono |
| (fun γ hγ => hγ.2 midpointTime) |
| let v0 : ProbabilityMeasure X := ⟨ν₀, inferInstance⟩ |
| let v1 : ProbabilityMeasure X := ⟨ν₁, inferInstance⟩ |
| obtain ⟨q, hq, hgq, hpq, _⟩ := hcd ν₀ ν₁ inferInstance inferInstance |
| (Measure.absolutelyContinuous_of_le_smul h0) |
| (Measure.absolutelyContinuous_of_le_smul h1) hm0 hm1 |
| have hne : (OptimalGeodesicPlans v0 v1).Nonempty := ⟨⟨q, hq⟩, hgq, hpq⟩ |
| obtain ⟨π, hπ, hmπ⟩ := exists_optimal_geodesic_plan_minimizing_excess |
| (μ.restrict W) v0 v1 p hR hs0 hs1 hne midpointTime hC |
| have hmin : ∀ (q : Measure C(Time, X)), IsProbabilityMeasure q → |
| (∀ᵐ γ ∂q, IsGeodesic γ) → |
| IsOptimalCoupling ν₀ ν₁ (q.map (fun γ => (γ 0, γ 1))) → |
| transportExcessMass (μ.restrict W) ((π : Measure C(Time, X)).map (fun γ => γ midpointTime)) C ≤ |
| transportExcessMass (μ.restrict W) (q.map (fun γ => γ midpointTime)) C := by |
| intro q hq hgq hpq |
| exact hmπ ⟨q, hq⟩ ⟨hgq, hpq⟩ |
| have hD : ∀ x ∈ closedBall p R, ∀ y ∈ closedBall p R, dist x y ≤ 2 * R := by |
| intro x hx y hy |
| have hd := dist_triangle x p y |
| rw [dist_comm p y] at hd |
| change dist x p ≤ R at hx |
| change dist y p ≤ R at hy |
| linarith |
| have hz := hcd.localized_excess_minimizer_zero_asymmetric μ W hW K n ν₀ ν₁ hC₀ hC₁ hC hCpos |
| h0 h1 hm0 hm1 (closedBall p R) (closedBall p R) hs0 hs1 hn (2 * R) hD |
| hsupport hfill (π : Measure C(Time, X)) hπ.1 hπ.2 hmin |
| refine ⟨π, inferInstance, hπ.1, hπ.2, ?_⟩ |
| have hs := hsupport (π : Measure C(Time, X)) inferInstance hπ.1 hπ.2 |
| have ht := localized_excess_zero_iff μ ((π : Measure C(Time, X)).map (fun γ => γ midpointTime)) |
| (B := W) (C := C) hs hC |
| exact ht.mp hz |
|
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| end Attainment |
| end RegularPointCoordinates |
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| end OAI |
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