| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.Squared |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology Classical |
|
|
| variable {E : Type*} [MetricSpace E] [CompleteSpace E] [SecondCountableTopology E] |
| [MeasurableSpace E] [BorelSpace E] |
|
|
| def DistanceCoupled (μ ν : Measure E) (R : ℝ) : Prop := |
| ∃κ : Measure (E×E),IsProbabilityMeasure κ ∧ κ.fst=μ ∧ κ.snd=ν ∧ |
| ∀ᵐ z ∂κ,dist z.1 z.2≤R |
|
|
| lemma tight_range_of_tendsto {μ : ProbabilityMeasure E} {μs : ℕ → ProbabilityMeasure E} |
| (h : Tendsto μs atTop (𝓝 μ)) : IsTightMeasureSet (range (fun n => (μs n : Measure E))) := by |
| have hc : IsCompact (closure (range μs)) := h.isCompact_insert_range.of_isClosed_subset |
| isClosed_closure (closure_minimal (subset_insert _ _) h.isCompact_insert_range.isClosed) |
| have hh := isTightMeasureSet_of_isCompact_closure hc |
| convert hh using 1 |
| ext ν |
| simp only [mem_range,mem_ofPred_eq] |
| aesop |
|
|
| theorem distanceCoupled_of_tendsto {μ ν : ProbabilityMeasure E} |
| {μs νs : ℕ → ProbabilityMeasure E} (hμ : Tendsto μs atTop (𝓝 μ)) |
| (hν : Tendsto νs atTop (𝓝 ν)) (R : ℝ) |
| (hc : ∀n,DistanceCoupled (μs n : Measure E) (νs n : Measure E) R) : DistanceCoupled (μ : Measure E) (ν : Measure E) R := by |
| choose κ hκ hfst hsnd hdist using hc |
| let Ks : ℕ → ProbabilityMeasure (E×E) := fun n => ⟨κ n,hκ n⟩ |
| have hj : IsTightMeasureSet (range κ) := by |
| apply IsTightMeasureSet.prodMk |
| · have he : Measure.fst '' range κ=range (fun n => (μs n : Measure E)) := by |
| ext ρ |
| simp only [mem_image,mem_range,exists_exists_eq_and] |
| simp only [hfst] |
| rw [he] |
| exact tight_range_of_tendsto hμ |
| · have he : Measure.snd '' range κ=range (fun n => (νs n : Measure E)) := by |
| ext ρ |
| simp only [mem_image,mem_range,exists_exists_eq_and] |
| simp only [hsnd] |
| rw [he] |
| exact tight_range_of_tendsto hν |
| have hj' : IsTightMeasureSet {((ρ : ProbabilityMeasure (E×E)) : Measure (E×E)) | ρ∈range Ks} := by |
| convert hj using 1 |
| ext ρ |
| simp only [mem_ofPred_eq,mem_range,exists_exists_eq_and] |
| rfl |
| obtain ⟨K,_,φ,hφ,hK⟩ := (isCompact_closure_of_isTightMeasureSet hj').tendsto_subseq |
| (fun n => subset_closure (mem_range_self n)) |
| have hf := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous (Ks ∘ φ) K hK continuous_fst |
| have hg := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous (Ks ∘ φ) K hK continuous_snd |
| have hfm : (fun n => (Ks (φ n)).map Prod.fst)=μs ∘ φ := by |
| funext n; apply Subtype.ext; exact hfst (φ n) |
| have hgm : (fun n => (Ks (φ n)).map Prod.snd)=νs ∘ φ := by |
| funext n; apply Subtype.ext; exact hsnd (φ n) |
| simp only [Function.comp_apply] at hf hg |
| rw [hfm] at hf |
| rw [hgm] at hg |
| have hef := tendsto_nhds_unique hf (hμ.comp hφ.tendsto_atTop) |
| have heg := tendsto_nhds_unique hg (hν.comp hφ.tendsto_atTop) |
| refine ⟨K,inferInstance,congrArg (fun p : ProbabilityMeasure E => (p : Measure E)) hef, |
| congrArg (fun p : ProbabilityMeasure E => (p : Measure E)) heg,?_⟩ |
| let S : Set (E×E) := {z | dist z.1 z.2≤R} |
| have hS : IsClosed S := isClosed_le (continuous_fst.dist continuous_snd) continuous_const |
| have hprob (n : ℕ) : (Ks n : Measure (E×E)) S=1 := by |
| calc (Ks n : Measure (E×E)) S = (Ks n : Measure (E×E)) univ := by |
| apply measure_congr |
| filter_upwards [hdist n] with z hz |
| apply propext |
| change (dist z.1 z.2≤R) ↔ True |
| exact iff_true_intro hz |
| _ = 1 := measure_univ |
| have hl := ProbabilityMeasure.limsup_measure_closed_le_of_tendsto hK hS |
| simp only [Function.comp_apply,hprob,limsup_const] at hl |
| have he : (K : Measure (E×E)) S=1 := le_antisymm (by simpa only [measure_univ] using (measure_mono (subset_univ S) : (K : Measure (E×E)) S≤(K : Measure (E×E)) univ)) hl |
| rw [←show (∀ᵐ z : E×E ∂(K : Measure (E×E)),z∈S) ↔ (∀ᵐ z ∂(K : Measure (E×E)),dist z.1 z.2≤R) from Iff.rfl] |
| exact (ae_iff).mpr (by |
| change (K : Measure (E×E)) Sᶜ=0 |
| rw [measure_compl hS.measurableSet (measure_ne_top _ _),measure_univ,he,tsub_self]) |
|
|
| lemma DistanceCoupled.squared {d : ℕ} {μ ν : Measure (Point d)} {R : ℝ} |
| (hr : 0≤R) (h : DistanceCoupled μ ν R) : SquaredAt μ ν id id (R^2) := by |
| obtain ⟨κ,hκ,hf,hg,hd⟩ := h |
| let := hκ |
| have hb : ∀ᵐ z ∂κ, ‖z.1-z.2‖^2≤R^2 := hd.mono (fun z hz => |
| (sq_le_sq₀ (norm_nonneg _) hr).2 (by simpa only [dist_eq_norm] using hz)) |
| have hi : Integrable (fun z : Point d × Point d => ‖z.1-z.2‖^2) κ := |
| (integrable_const (R^2)).mono' (by fun_prop) |
| (hb.mono (fun z hz => by simpa only [Real.norm_eq_abs,abs_sq] using hz)) |
| refine ⟨κ,hκ,hf,hg,hi,?_⟩ |
| simpa only [id_eq,integral_const,probReal_univ,one_smul] using integral_mono_ae hi (integrable_const (R^2)) hb |
| end LogConcaveSampling.Coupling |
| |
| end |
| |
| end |
| |
| end OAI |
| |
