| import OAI.Analysis.Brenier.Basic | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace Problem358 | |
| variable {d : ℕ} {ρ : Measure (E d)} {T S U : E d → E d} | |
| @[simp] theorem mapL2Dist_self (ρ : Measure (E d)) (T : E d → E d) : | |
| mapL2Dist ρ T T = 0 := by | |
| simp [mapL2Dist] | |
| theorem mapL2Dist_nonneg (ρ : Measure (E d)) (T S : E d → E d) : | |
| 0 ≤ mapL2Dist ρ T S := Real.sqrt_nonneg _ | |
| theorem mapL2Dist_comm (ρ : Measure (E d)) (T S : E d → E d) : | |
| mapL2Dist ρ T S = mapL2Dist ρ S T := by | |
| simp only [mapL2Dist, norm_sub_rev (T _) (S _)] | |
| theorem mapL2Dist_congr_ae {T' S' : E d → E d} | |
| (hT : T =ᵐ[ρ] T') (hS : S =ᵐ[ρ] S') : | |
| mapL2Dist ρ T S = mapL2Dist ρ T' S' := by | |
| unfold mapL2Dist | |
| congr 1 | |
| apply integral_congr_ae | |
| filter_upwards [hT, hS] with x hx hy | |
| rw [hx, hy] | |
| theorem mapL2Dist_eq_zero_of_ae_eq (h : T =ᵐ[ρ] S) : | |
| mapL2Dist ρ T S = 0 := by | |
| rw [mapL2Dist_congr_ae h (Filter.EventuallyEq.refl _ _), mapL2Dist_self] | |
| theorem mapL2Dist_sq (ρ : Measure (E d)) (T S : E d → E d) : | |
| mapL2Dist ρ T S ^ 2 = ∫ x, ‖T x - S x‖ ^ 2 ∂ρ := by | |
| exact Real.sq_sqrt (integral_nonneg fun x => sq_nonneg _) | |
| theorem mapL2Dist_eq_zero_iff | |
| (hInt : Integrable (fun x => ‖T x - S x‖ ^ 2) ρ) : | |
| mapL2Dist ρ T S = 0 ↔ T =ᵐ[ρ] S := by | |
| constructor | |
| · intro h | |
| have hIntegral : (∫ x, ‖T x - S x‖ ^ 2 ∂ρ) = 0 := by | |
| rw [← mapL2Dist_sq, h] | |
| simp | |
| have ha := (integral_eq_zero_iff_of_nonneg (fun x => sq_nonneg _) hInt).mp hIntegral | |
| filter_upwards [ha] with x hx | |
| simpa only [Pi.zero_apply, sq_eq_zero_iff, norm_eq_zero, sub_eq_zero] using hx | |
| · exact mapL2Dist_eq_zero_of_ae_eq | |
| theorem quadraticPlanCost_map_pair (hT : Measurable T) (hS : Measurable S) : | |
| quadraticPlanCost (Measure.map (fun x => (T x, S x)) ρ) = | |
| ∫⁻ x, ENNReal.ofReal (‖T x - S x‖ ^ 2) ∂ρ := by | |
| unfold quadraticPlanCost | |
| exact lintegral_map (by fun_prop) (hT.prodMk hS) | |
| theorem mapL2Dist_eq_sqrt_pair_cost (hT : Measurable T) (hS : Measurable S) : | |
| mapL2Dist ρ T S = Real.sqrt | |
| (quadraticPlanCost (Measure.map (fun x => (T x, S x)) ρ)).toReal := by | |
| rw [mapL2Dist, quadraticPlanCost_map_pair hT hS] | |
| congr 1 | |
| apply integral_eq_lintegral_of_nonneg_ae | |
| · exact ae_of_all _ fun x => sq_nonneg _ | |
| · fun_prop | |
| theorem mapL2Dist_eq_lpNorm (hT : AEStronglyMeasurable T ρ) | |
| (hS : AEStronglyMeasurable S ρ) : | |
| mapL2Dist ρ T S = lpNorm (T - S) 2 ρ := by | |
| rw [lpNorm_eq_integral_norm_rpow_toReal (by norm_num) (by norm_num) (hT.sub hS)] | |
| simp [mapL2Dist, Real.sqrt_eq_rpow] | |
| theorem mapL2Dist_eq_dist_toLp (hT : MemLp T 2 ρ) (hS : MemLp S 2 ρ) : | |
| mapL2Dist ρ T S = dist (hT.toLp T) (hS.toLp S) := by | |
| rw [Lp.dist_edist, Lp.edist_toLp_toLp, toReal_eLpNorm] | |
| exact mapL2Dist_eq_lpNorm hT.aestronglyMeasurable hS.aestronglyMeasurable | |
| theorem mapL2Dist_triangle (hT : MemLp T 2 ρ) (hS : MemLp S 2 ρ) | |
| (hU : MemLp U 2 ρ) : | |
| mapL2Dist ρ T U ≤ mapL2Dist ρ T S + mapL2Dist ρ S U := by | |
| rw [mapL2Dist_eq_dist_toLp hT hU, mapL2Dist_eq_dist_toLp hT hS, | |
| mapL2Dist_eq_dist_toLp hS hU] | |
| exact dist_triangle _ _ _ | |
| end Problem358 | |
| namespace Problem358 | |
| variable {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V] | |
| [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] | |
| /-- A bounded nonnegative squared separation is integrable against a finite measure. -/ | |
| theorem integrable_norm_sub_sq_of_ae_bound {ρ : Measure X} [IsFiniteMeasure ρ] | |
| {T S : X → V} (hT : Measurable T) (hS : Measurable S) | |
| {R : ℝ} (hbound : ∀ᵐ x ∂ρ, ‖T x - S x‖ ≤ R) : | |
| Integrable (fun x => ‖T x - S x‖ ^ 2) ρ := by | |
| refine (integrable_const (R ^ 2)).mono' ?_ ?_ | |
| · exact ((hT.sub hS).norm.pow_const 2).aestronglyMeasurable | |
| · filter_upwards [hbound] with x hx | |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] | |
| exact pow_le_pow_left₀ (norm_nonneg _) hx 2 | |
| omit [NormedAddCommGroup V] [BorelSpace V] [SecondCountableTopology V] in | |
| /-- Support of a pushforward pulls back to an almost-everywhere range condition. -/ | |
| theorem ae_mem_of_map_compl_eq_zero {ρ : Measure X} {T : X → V} | |
| (hT : Measurable T) {Y : Set V} (hY : (Measure.map T ρ) Yᶜ = 0) : | |
| ∀ᵐ x ∂ρ, T x ∈ Y := by | |
| apply ae_of_ae_map hT.aemeasurable | |
| exact ae_iff.mpr hY | |
| /-- Two measurable maps with essentially bounded ranges have integrable squared separation. -/ | |
| theorem integrable_norm_sub_sq_of_ae_mem_bounded {ρ : Measure X} [IsFiniteMeasure ρ] | |
| {T S : X → V} (hT : Measurable T) (hS : Measurable S) | |
| {Y Z : Set V} (hY : Bornology.IsBounded Y) (hZ : Bornology.IsBounded Z) | |
| (hTY : ∀ᵐ x ∂ρ, T x ∈ Y) (hSZ : ∀ᵐ x ∂ρ, S x ∈ Z) : | |
| Integrable (fun x => ‖T x - S x‖ ^ 2) ρ := by | |
| obtain ⟨R, hR⟩ := hY.exists_norm_le | |
| obtain ⟨Q, hQ⟩ := hZ.exists_norm_le | |
| apply integrable_norm_sub_sq_of_ae_bound hT hS (R := R + Q) | |
| filter_upwards [hTY, hSZ] with x hx hy | |
| exact (norm_sub_le _ _).trans (add_le_add (hR _ hx) (hQ _ hy)) | |
| /-- Compact support of the two pushforwards suffices for the L² separation integral. -/ | |
| theorem integrable_norm_sub_sq_of_map_compact {ρ : Measure X} [IsFiniteMeasure ρ] | |
| {T S : X → V} (hT : Measurable T) (hS : Measurable S) | |
| {Y Z : Set V} (hY : IsCompact Y) (hZ : IsCompact Z) | |
| (hTY : (Measure.map T ρ) Yᶜ = 0) (hSZ : (Measure.map S ρ) Zᶜ = 0) : | |
| Integrable (fun x => ‖T x - S x‖ ^ 2) ρ := | |
| integrable_norm_sub_sq_of_ae_mem_bounded hT hS hY.isBounded hZ.isBounded | |
| (ae_mem_of_map_compl_eq_zero hT hTY) (ae_mem_of_map_compl_eq_zero hS hSZ) | |
| omit [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] in | |
| /-- The squared L² separation integral vanishes exactly for a.e.-equal maps. -/ | |
| theorem integral_norm_sub_sq_eq_zero_iff {ρ : Measure X} {T S : X → V} | |
| (hTS : Integrable (fun x => ‖T x - S x‖ ^ 2) ρ) : | |
| (∫ x, ‖T x - S x‖ ^ 2 ∂ρ) = 0 ↔ T =ᵐ[ρ] S := by | |
| rw [integral_eq_zero_iff_of_nonneg (fun x => sq_nonneg _) hTS] | |
| simp only [Filter.EventuallyEq, Pi.zero_apply, sq_eq_zero_iff, norm_eq_zero, sub_eq_zero] | |
| end Problem358 | |
| end | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology ENNReal | |
| namespace Problem358 | |
| variable {d : ℕ} {ρ : Measure (E d)} | |
| theorem exists_mapL2Dist_limit {T : ℕ → E d → E d} | |
| (hT : ∀ n, MemLp (T n) 2 ρ) | |
| (hC : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, | |
| mapL2Dist ρ (T m) (T n) < ε) : | |
| ∃ S : E d → E d, Measurable S ∧ MemLp S 2 ρ ∧ | |
| Tendsto (fun n => mapL2Dist ρ (T n) S) atTop (𝓝 0) := by | |
| let F : ℕ → Lp (E d) 2 ρ := fun n => (hT n).toLp (T n) | |
| have hCF : CauchySeq F := by | |
| apply Metric.cauchySeq_iff.mpr | |
| intro ε hε | |
| obtain ⟨N, hN⟩ := hC ε hε | |
| refine ⟨N, fun m hm n hn => ?_⟩ | |
| simpa only [F, ← mapL2Dist_eq_dist_toLp] using hN m hm n hn | |
| obtain ⟨S, hS⟩ := cauchySeq_tendsto_of_complete hCF | |
| refine ⟨S, (Lp.stronglyMeasurable S).measurable, (Lp.memLp S), ?_⟩ | |
| have hDist := tendsto_iff_dist_tendsto_zero.mp hS | |
| simpa only [mapL2Dist_eq_dist_toLp (hT _) (Lp.memLp S), Lp.toLp_coeFn, F] | |
| using hDist | |
| theorem exists_subseq_tendsto_ae_of_mapL2Dist_tendsto_zero | |
| {T : ℕ → E d → E d} {S : E d → E d} | |
| (hT : ∀ n, MemLp (T n) 2 ρ) (hS : MemLp S 2 ρ) | |
| (hlim : Tendsto (fun n => mapL2Dist ρ (T n) S) atTop (𝓝 0)) : | |
| ∃ ns : ℕ → ℕ, StrictMono ns ∧ | |
| ∀ᵐ x ∂ρ, Tendsto (fun n => T (ns n) x) atTop (𝓝 (S x)) := by | |
| let F : ℕ → Lp (E d) 2 ρ := fun n => (hT n).toLp (T n) | |
| have hLp : Tendsto F atTop (𝓝 (hS.toLp S)) := by | |
| apply tendsto_iff_dist_tendsto_zero.mpr | |
| simpa only [F, ← mapL2Dist_eq_dist_toLp] using hlim | |
| obtain ⟨ns, hns, hae⟩ := | |
| (tendstoInMeasure_of_tendsto_Lp hLp).exists_seq_tendsto_ae | |
| refine ⟨ns, hns, ?_⟩ | |
| have hEq : ∀ᵐ x ∂ρ, ∀ n, F n x = T n x := by | |
| exact ae_all_iff.mpr (fun n => (hT n).coeFn_toLp) | |
| filter_upwards [hae, hEq, hS.coeFn_toLp] with x hx hTx hSx | |
| simpa only [hTx, hSx] using hx | |
| theorem ae_mem_of_mapL2Dist_tendsto_zero | |
| {T : ℕ → E d → E d} {S : E d → E d} {Y : Set (E d)} | |
| (hT : ∀ n, MemLp (T n) 2 ρ) (hS : MemLp S 2 ρ) | |
| (hlim : Tendsto (fun n => mapL2Dist ρ (T n) S) atTop (𝓝 0)) | |
| (hY : IsClosed Y) (hmem : ∀ n, ∀ᵐ x ∂ρ, T n x ∈ Y) : | |
| ∀ᵐ x ∂ρ, S x ∈ Y := by | |
| obtain ⟨ns, _, hns⟩ := | |
| exists_subseq_tendsto_ae_of_mapL2Dist_tendsto_zero hT hS hlim | |
| have hAll : ∀ᵐ x ∂ρ, ∀ n, T n x ∈ Y := ae_all_iff.mpr hmem | |
| filter_upwards [hns, hAll] with x hx hTx | |
| exact hY.mem_of_tendsto hx (Eventually.of_forall fun n => hTx (ns n)) | |
| theorem exists_mapL2Dist_limit_in_closed {T : ℕ → E d → E d} {Y : Set (E d)} | |
| (hT : ∀ n, MemLp (T n) 2 ρ) | |
| (hC : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, | |
| mapL2Dist ρ (T m) (T n) < ε) | |
| (hY : IsClosed Y) (hmem : ∀ n, ∀ᵐ x ∂ρ, T n x ∈ Y) : | |
| ∃ S : E d → E d, Measurable S ∧ MemLp S 2 ρ ∧ | |
| (∀ᵐ x ∂ρ, S x ∈ Y) ∧ | |
| Tendsto (fun n => mapL2Dist ρ (T n) S) atTop (𝓝 0) := by | |
| obtain ⟨S, hSm, hS, hlim⟩ := exists_mapL2Dist_limit hT hC | |
| exact ⟨S, hSm, hS, ae_mem_of_mapL2Dist_tendsto_zero hT hS hlim hY hmem, hlim⟩ | |
| /-- Map distance is continuous under L² convergence of each endpoint. -/ | |
| theorem tendsto_mapL2Dist_of_tendsto {ι : Type*} {l : Filter ι} | |
| {T S : ι → E d → E d} {T₀ S₀ : E d → E d} | |
| (hT : ∀ n, MemLp (T n) 2 ρ) (hS : ∀ n, MemLp (S n) 2 ρ) | |
| (hT₀ : MemLp T₀ 2 ρ) (hS₀ : MemLp S₀ 2 ρ) | |
| (hTlim : Tendsto (fun n => mapL2Dist ρ (T n) T₀) l (𝓝 0)) | |
| (hSlim : Tendsto (fun n => mapL2Dist ρ (S n) S₀) l (𝓝 0)) : | |
| Tendsto (fun n => mapL2Dist ρ (T n) (S n)) l (𝓝 (mapL2Dist ρ T₀ S₀)) := by | |
| have hTLp : Tendsto (fun n => (hT n).toLp (T n)) l (𝓝 (hT₀.toLp T₀)) := by | |
| apply tendsto_iff_dist_tendsto_zero.mpr | |
| simpa only [← mapL2Dist_eq_dist_toLp] using hTlim | |
| have hSLp : Tendsto (fun n => (hS n).toLp (S n)) l (𝓝 (hS₀.toLp S₀)) := by | |
| apply tendsto_iff_dist_tendsto_zero.mpr | |
| simpa only [← mapL2Dist_eq_dist_toLp] using hSlim | |
| simpa only [← mapL2Dist_eq_dist_toLp] using hTLp.dist hSLp | |
| end Problem358 | |
| end | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem358 | |
| /-- The graph plan of a measurable map remembers its almost-everywhere class. -/ | |
| theorem graphPlan_eq_iff_ae_eq {d : ℕ} {ρ : Measure (E d)} | |
| {T S : E d → E d} (hT : Measurable T) (hS : Measurable S) : | |
| graphPlan ρ T = graphPlan ρ S ↔ T =ᵐ[ρ] S := by | |
| constructor | |
| · intro heq | |
| have hgraph : ∀ᵐ p ∂graphPlan ρ T, p.2 = T p.1 := by | |
| apply (ae_map_iff (measurable_id.prodMk hT).aemeasurable | |
| (measurableSet_eq_fun measurable_snd (hT.comp measurable_fst))).2 | |
| exact Filter.Eventually.of_forall fun _ => rfl | |
| rw [heq] at hgraph | |
| have hae := ae_of_ae_map (measurable_id.prodMk hS).aemeasurable hgraph | |
| filter_upwards [hae] with x hx | |
| exact hx.symm | |
| · intro hae | |
| apply Measure.map_congr | |
| filter_upwards [hae] with x hx | |
| exact Prod.ext rfl hx | |
| /-- A measurable transport of a probability source defines a coupling with its law. -/ | |
| theorem graphPlan_isCoupling {d : ℕ} {ρ μ : Measure (E d)} | |
| [IsProbabilityMeasure ρ] {T : E d → E d} | |
| (hT : Measurable T) (hlaw : Measure.map T ρ = μ) : | |
| IsCoupling ρ μ (graphPlan ρ T) := by | |
| have hm : Measurable (fun x => (x, T x)) := measurable_id.prodMk hT | |
| refine ⟨?_, ?_, ?_⟩ | |
| · unfold graphPlan | |
| infer_instance | |
| · rw [graphPlan, Measure.map_map measurable_fst hm] | |
| exact Measure.map_id | |
| · rw [graphPlan, Measure.map_map measurable_snd hm] | |
| exact hlaw | |
| /-- All optimal maps for the same source and target have the same graph cost. -/ | |
| theorem IsQuadraticOptimalMap.cost_eq {d : ℕ} {ρ μ : Measure (E d)} | |
| [IsProbabilityMeasure ρ] {T S : E d → E d} | |
| (hT : IsQuadraticOptimalMap ρ μ T) (hS : IsQuadraticOptimalMap ρ μ S) : | |
| quadraticPlanCost (graphPlan ρ T) = quadraticPlanCost (graphPlan ρ S) := by | |
| apply le_antisymm | |
| · exact hT.2.2 _ (graphPlan_isCoupling hS.1 hS.2.1) | |
| · exact hS.2.2 _ (graphPlan_isCoupling hT.1 hT.2.1) | |
| /-- Uniqueness among transport plans implies uniqueness of the map almost everywhere. -/ | |
| theorem IsUniqueQuadraticOptimalMap.ae_eq_of_optimal {d : ℕ} | |
| {ρ μ : Measure (E d)} [IsProbabilityMeasure ρ] {T S : E d → E d} | |
| (hT : IsUniqueQuadraticOptimalMap ρ μ T) | |
| (hS : IsQuadraticOptimalMap ρ μ S) : T =ᵐ[ρ] S := by | |
| apply (graphPlan_eq_iff_ae_eq hT.1.1 hS.1).mp | |
| exact (hT.2 _ (graphPlan_isCoupling hS.1 hS.2.1) | |
| (hS.2.2 _ (graphPlan_isCoupling hT.1.1 hT.1.2.1))).symm | |
| /-- Distances between unique optimal maps do not depend on selected representatives. -/ | |
| theorem mapL2Dist_eq_of_optimal_maps {d : ℕ} {ρ μ ν : Measure (E d)} | |
| [IsProbabilityMeasure ρ] {T S T' S' : E d → E d} | |
| (hT : IsUniqueQuadraticOptimalMap ρ μ T) | |
| (hS : IsUniqueQuadraticOptimalMap ρ ν S) | |
| (hT' : IsQuadraticOptimalMap ρ μ T') (hS' : IsQuadraticOptimalMap ρ ν S') : | |
| mapL2Dist ρ T S = mapL2Dist ρ T' S' := | |
| mapL2Dist_congr_ae (hT.ae_eq_of_optimal hT') (hS.ae_eq_of_optimal hS') | |
| /-- Changing a measurable representative on a null set preserves optimality. -/ | |
| theorem IsQuadraticOptimalMap.congr_ae {d : ℕ} {ρ μ : Measure (E d)} | |
| {T S : E d → E d} (hT : IsQuadraticOptimalMap ρ μ T) | |
| (hS : Measurable S) (hae : T =ᵐ[ρ] S) : IsQuadraticOptimalMap ρ μ S := by | |
| have hg := (graphPlan_eq_iff_ae_eq hT.1 hS).mpr hae | |
| refine ⟨hS, ?_, ?_⟩ | |
| · exact (Measure.map_congr hae).symm.trans hT.2.1 | |
| · intro π hπ | |
| rw [← hg] | |
| exact hT.2.2 π hπ | |
| /-- Changing a measurable representative on a null set also preserves plan uniqueness. -/ | |
| theorem IsUniqueQuadraticOptimalMap.congr_ae {d : ℕ} {ρ μ : Measure (E d)} | |
| {T S : E d → E d} (hT : IsUniqueQuadraticOptimalMap ρ μ T) | |
| (hS : Measurable S) (hae : T =ᵐ[ρ] S) : | |
| IsUniqueQuadraticOptimalMap ρ μ S := by | |
| have hg := (graphPlan_eq_iff_ae_eq hT.1.1 hS).mpr hae | |
| refine ⟨hT.1.congr_ae hS hae, ?_⟩ | |
| intro π hπ hcost | |
| rw [← hg] at hcost ⊢ | |
| exact hT.2 π hπ hcost | |
| end Problem358 | |
| end | |
| noncomputable section | |
| open Filter | |
| open scoped Topology | |
| namespace Problem358.StabilityLimit | |
| /-- A uniform power bound survives convergence when the input distances have only | |
| an asymptotic upper bound. This is enough for finite-target quantization. -/ | |
| theorem real_bound_of_approximation | |
| {D W C α : ℝ} {Dₙ Wₙ ε : ℕ → ℝ} | |
| (hC : 0 ≤ C) (hα : 0 ≤ α) (hWₙ : ∀ n, 0 ≤ Wₙ n) | |
| (hD : Tendsto Dₙ atTop (𝓝 D)) (hε : Tendsto ε atTop (𝓝 0)) | |
| (hupper : ∀ n, Wₙ n ≤ W + ε n) | |
| (hbound : ∀ n, Dₙ n ≤ C * Real.rpow (Wₙ n) α) : | |
| D ≤ C * Real.rpow W α := by | |
| have hinput : Tendsto (fun n => W + ε n) atTop (𝓝 W) := by | |
| simpa using tendsto_const_nhds.add hε | |
| have hpower : Tendsto (fun n => C * Real.rpow (W + ε n) α) | |
| atTop (𝓝 (C * Real.rpow W α)) := | |
| tendsto_const_nhds.mul ((Real.continuous_rpow_const hα).tendsto W |>.comp hinput) | |
| apply le_of_tendsto_of_tendsto hD hpower | |
| exact Filter.Eventually.of_forall fun n => (hbound n).trans | |
| (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (hWₙ n) (hupper n) hα) hC) | |
| /-- Metric version of the approximation closure step. -/ | |
| theorem dist_bound_of_approximation {X : Type*} [PseudoMetricSpace X] | |
| {x y : X} {xₙ yₙ : ℕ → X} {W C α : ℝ} {Wₙ ε : ℕ → ℝ} | |
| (hC : 0 ≤ C) (hα : 0 ≤ α) (hWₙ : ∀ n, 0 ≤ Wₙ n) | |
| (hx : Tendsto xₙ atTop (𝓝 x)) (hy : Tendsto yₙ atTop (𝓝 y)) | |
| (hε : Tendsto ε atTop (𝓝 0)) (hupper : ∀ n, Wₙ n ≤ W + ε n) | |
| (hbound : ∀ n, dist (xₙ n) (yₙ n) ≤ C * Real.rpow (Wₙ n) α) : | |
| dist x y ≤ C * Real.rpow W α := by | |
| exact real_bound_of_approximation hC hα hWₙ (hx.dist hy) hε hupper hbound | |
| /-- The exact exponent used by the stability target. -/ | |
| theorem one_third_dist_bound_of_approximation {X : Type*} [PseudoMetricSpace X] | |
| {x y : X} {xₙ yₙ : ℕ → X} {W C : ℝ} {Wₙ ε : ℕ → ℝ} | |
| (hC : 0 ≤ C) (hWₙ : ∀ n, 0 ≤ Wₙ n) | |
| (hx : Tendsto xₙ atTop (𝓝 x)) (hy : Tendsto yₙ atTop (𝓝 y)) | |
| (hε : Tendsto ε atTop (𝓝 0)) (hupper : ∀ n, Wₙ n ≤ W + ε n) | |
| (hbound : ∀ n, dist (xₙ n) (yₙ n) ≤ C * Real.rpow (Wₙ n) (1 / 3 : ℝ)) : | |
| dist x y ≤ C * Real.rpow W (1 / 3 : ℝ) := by | |
| exact dist_bound_of_approximation hC (by norm_num) hWₙ hx hy hε hupper hbound | |
| open MeasureTheory | |
| variable {d : ℕ} {ρ : Measure (E d)} | |
| /-- Strong L² convergence in both arguments gives convergence of the map distance. -/ | |
| theorem tendsto_mapL2Dist {T S : E d → E d} {Tₙ Sₙ : ℕ → E d → E d} | |
| (hT : MemLp T 2 ρ) (hS : MemLp S 2 ρ) | |
| (hTₙ : ∀ n, MemLp (Tₙ n) 2 ρ) (hSₙ : ∀ n, MemLp (Sₙ n) 2 ρ) | |
| (hlimT : Tendsto (fun n => mapL2Dist ρ (Tₙ n) T) atTop (𝓝 0)) | |
| (hlimS : Tendsto (fun n => mapL2Dist ρ (Sₙ n) S) atTop (𝓝 0)) : | |
| Tendsto (fun n => mapL2Dist ρ (Tₙ n) (Sₙ n)) atTop (𝓝 (mapL2Dist ρ T S)) := by | |
| have hTLp : Tendsto (fun n => (hTₙ n).toLp (Tₙ n)) atTop (𝓝 (hT.toLp T)) := by | |
| apply tendsto_iff_dist_tendsto_zero.mpr | |
| simpa only [← mapL2Dist_eq_dist_toLp] using hlimT | |
| have hSLp : Tendsto (fun n => (hSₙ n).toLp (Sₙ n)) atTop (𝓝 (hS.toLp S)) := by | |
| apply tendsto_iff_dist_tendsto_zero.mpr | |
| simpa only [← mapL2Dist_eq_dist_toLp] using hlimS | |
| simpa only [← mapL2Dist_eq_dist_toLp] using hTLp.dist hSLp | |
| /-- Passage from finite-target map bounds to a strong-L² limiting pair. -/ | |
| theorem mapL2Dist_bound_of_approximation | |
| {T S : E d → E d} {Tₙ Sₙ : ℕ → E d → E d} | |
| {W C α : ℝ} {Wₙ ε : ℕ → ℝ} | |
| (hT : MemLp T 2 ρ) (hS : MemLp S 2 ρ) | |
| (hTₙ : ∀ n, MemLp (Tₙ n) 2 ρ) (hSₙ : ∀ n, MemLp (Sₙ n) 2 ρ) | |
| (hlimT : Tendsto (fun n => mapL2Dist ρ (Tₙ n) T) atTop (𝓝 0)) | |
| (hlimS : Tendsto (fun n => mapL2Dist ρ (Sₙ n) S) atTop (𝓝 0)) | |
| (hC : 0 ≤ C) (hα : 0 ≤ α) (hWₙ : ∀ n, 0 ≤ Wₙ n) | |
| (hε : Tendsto ε atTop (𝓝 0)) (hupper : ∀ n, Wₙ n ≤ W + ε n) | |
| (hbound : ∀ n, mapL2Dist ρ (Tₙ n) (Sₙ n) ≤ C * Real.rpow (Wₙ n) α) : | |
| mapL2Dist ρ T S ≤ C * Real.rpow W α := by | |
| exact real_bound_of_approximation hC hα hWₙ | |
| (tendsto_mapL2Dist hT hS hTₙ hSₙ hlimT hlimS) hε hupper hbound | |
| /-- Final one-third estimate under explicit finite approximation hypotheses. -/ | |
| theorem one_third_bound_of_target_approximation | |
| {T S : E d → E d} {Tₙ Sₙ : ℕ → E d → E d} | |
| {μ ν : Measure (E d)} {μₙ νₙ : ℕ → Measure (E d)} {C : ℝ} {ε : ℕ → ℝ} | |
| (hT : MemLp T 2 ρ) (hS : MemLp S 2 ρ) | |
| (hTₙ : ∀ n, MemLp (Tₙ n) 2 ρ) (hSₙ : ∀ n, MemLp (Sₙ n) 2 ρ) | |
| (hlimT : Tendsto (fun n => mapL2Dist ρ (Tₙ n) T) atTop (𝓝 0)) | |
| (hlimS : Tendsto (fun n => mapL2Dist ρ (Sₙ n) S) atTop (𝓝 0)) | |
| (hC : 0 ≤ C) (hε : Tendsto ε atTop (𝓝 0)) | |
| (hupper : ∀ n, wasserstein2 (μₙ n) (νₙ n) ≤ wasserstein2 μ ν + ε n) | |
| (hbound : ∀ n, mapL2Dist ρ (Tₙ n) (Sₙ n) ≤ | |
| C * Real.rpow (wasserstein2 (μₙ n) (νₙ n)) (1 / 3 : ℝ)) : | |
| mapL2Dist ρ T S ≤ C * Real.rpow (wasserstein2 μ ν) (1 / 3 : ℝ) := by | |
| exact mapL2Dist_bound_of_approximation hT hS hTₙ hSₙ hlimT hlimS hC | |
| (by norm_num) (fun n => Real.sqrt_nonneg _) hε hupper hbound | |
| end Problem358.StabilityLimit | |
| end | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem358 | |
| /-- On a probability source, an essential pointwise distance bound is an L² bound. -/ | |
| theorem mapL2Dist_le_of_ae_bound {d : ℕ} {ρ : Measure (E d)} | |
| [IsProbabilityMeasure ρ] {T S : E d → E d} | |
| (hT : Measurable T) (hS : Measurable S) {M : ℝ} (hM : 0 ≤ M) | |
| (hbound : ∀ᵐ x ∂ρ, ‖T x - S x‖ ≤ M) : | |
| mapL2Dist ρ T S ≤ M := by | |
| apply (Real.sqrt_le_left hM).mpr | |
| have hi := integrable_norm_sub_sq_of_ae_bound hT hS hbound | |
| have hle : (∫ x, ‖T x - S x‖ ^ 2 ∂ρ) ≤ ∫ _x : E d, M ^ 2 ∂ρ := by | |
| apply integral_mono_ae hi (integrable_const _) | |
| filter_upwards [hbound] with x hx | |
| exact pow_le_pow_left₀ (norm_nonneg _) hx 2 | |
| simpa using hle | |
| /-- Uniform norm bounds on two maps give a uniform L² distance bound. -/ | |
| theorem mapL2Dist_le_of_ae_norm_bounds {d : ℕ} {ρ : Measure (E d)} | |
| [IsProbabilityMeasure ρ] {T S : E d → E d} | |
| (hT : Measurable T) (hS : Measurable S) {R Q : ℝ} | |
| (hR : 0 ≤ R) (hQ : 0 ≤ Q) | |
| (hRT : ∀ᵐ x ∂ρ, ‖T x‖ ≤ R) (hQS : ∀ᵐ x ∂ρ, ‖S x‖ ≤ Q) : | |
| mapL2Dist ρ T S ≤ R + Q := by | |
| apply mapL2Dist_le_of_ae_bound hT hS (add_nonneg hR hQ) | |
| filter_upwards [hRT, hQS] with x hx hy | |
| exact (norm_sub_le _ _).trans (add_le_add hx hy) | |
| /-- The norms of optimal maps are controlled by a bound on the target supports. -/ | |
| theorem IsQuadraticOptimalMap.mapL2Dist_le_of_norm_bound {d : ℕ} | |
| {ρ μ ν : Measure (E d)} [IsProbabilityMeasure ρ] | |
| {T S : E d → E d} | |
| (hT : IsQuadraticOptimalMap ρ μ T) (hS : IsQuadraticOptimalMap ρ ν S) | |
| {Y : Set (E d)} (hμ : IsSupported μ Y) (hν : IsSupported ν Y) | |
| {L : ℝ} (hL : 0 ≤ L) (hY : ∀ y ∈ Y, ‖y‖ ≤ L) : | |
| mapL2Dist ρ T S ≤ 2 * L := by | |
| have hTY : ∀ᵐ x ∂ρ, T x ∈ Y := | |
| ae_mem_of_map_compl_eq_zero hT.1 (by simpa [hT.2.1, IsSupported] using hμ) | |
| have hSY : ∀ᵐ x ∂ρ, S x ∈ Y := | |
| ae_mem_of_map_compl_eq_zero hS.1 (by simpa [hS.2.1, IsSupported] using hν) | |
| have hRT : ∀ᵐ x ∂ρ, ‖T x‖ ≤ L := hTY.mono fun x hx => hY _ hx | |
| have hRS : ∀ᵐ x ∂ρ, ‖S x‖ ≤ L := hSY.mono fun x hx => hY _ hx | |
| simpa [two_mul] using mapL2Dist_le_of_ae_norm_bounds hT.1 hS.1 hL hL hRT hRS | |
| end Problem358 | |
| namespace Problem358 | |
| /-- Compact support of the pushforwards suffices for separation by map distance. -/ | |
| theorem mapL2Dist_eq_zero_iff_ae_of_compact {d : ℕ} {ρ : Measure (E d)} | |
| [IsFiniteMeasure ρ] {T S : E d → E d} (hT : Measurable T) (hS : Measurable S) | |
| {Y Z : Set (E d)} (hY : IsCompact Y) (hZ : IsCompact Z) | |
| (hTY : IsSupported (Measure.map T ρ) Y) (hSZ : IsSupported (Measure.map S ρ) Z) : | |
| mapL2Dist ρ T S = 0 ↔ T =ᵐ[ρ] S := | |
| mapL2Dist_eq_zero_iff (integrable_norm_sub_sq_of_map_compact hT hS hY hZ hTY hSZ) | |
| /-- Positive map distance is equivalent to distinct a.e. classes under integrability. -/ | |
| theorem mapL2Dist_pos_iff_not_ae {d : ℕ} {ρ : Measure (E d)} | |
| {T S : E d → E d} (hTS : Integrable (fun x => ‖T x - S x‖ ^ 2) ρ) : | |
| 0 < mapL2Dist ρ T S ↔ ¬ T =ᵐ[ρ] S := by | |
| rw [lt_iff_le_and_ne] | |
| simp only [mapL2Dist_nonneg, true_and, ne_eq, eq_comm (a := (0 : ℝ)), | |
| mapL2Dist_eq_zero_iff hTS] | |
| /-- Two unequal target laws force positive map distance for compact-supported maps. -/ | |
| theorem mapL2Dist_pos_of_map_ne_of_compact {d : ℕ} {ρ : Measure (E d)} | |
| [IsFiniteMeasure ρ] {T S : E d → E d} (hT : Measurable T) (hS : Measurable S) | |
| {Y Z : Set (E d)} (hY : IsCompact Y) (hZ : IsCompact Z) | |
| (hTY : IsSupported (Measure.map T ρ) Y) (hSZ : IsSupported (Measure.map S ρ) Z) | |
| (hne : Measure.map T ρ ≠ Measure.map S ρ) : 0 < mapL2Dist ρ T S := by | |
| refine lt_of_le_of_ne (mapL2Dist_nonneg ρ T S) ?_ | |
| intro hzero | |
| have hae : T =ᵐ[ρ] S := | |
| (mapL2Dist_eq_zero_iff_ae_of_compact hT hS hY hZ hTY hSZ).mp hzero.symm | |
| exact hne (Measure.map_congr hae) | |
| /-- Optimal maps to distinct compact-supported targets have positive map distance. -/ | |
| theorem IsQuadraticOptimalMap.mapL2Dist_pos_of_ne {d : ℕ} | |
| {ρ μ ν : Measure (E d)} [IsFiniteMeasure ρ] {T S : E d → E d} | |
| (hT : IsQuadraticOptimalMap ρ μ T) (hS : IsQuadraticOptimalMap ρ ν S) | |
| {Y Z : Set (E d)} (hY : IsCompact Y) (hZ : IsCompact Z) | |
| (hμ : IsSupported μ Y) (hν : IsSupported ν Z) (hne : μ ≠ ν) : | |
| 0 < mapL2Dist ρ T S := by | |
| apply mapL2Dist_pos_of_map_ne_of_compact hT.1 hS.1 hY hZ | |
| · simpa only [hT.2.1] using hμ | |
| · simpa only [hS.2.1] using hν | |
| · simpa only [hT.2.1, hS.2.1] using hne | |
| end Problem358 | |
| end | |
| end OAI | |