| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.Squared |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical |
|
|
| variable {Ω Γ W : Type*} [MeasurableSpace Ω] [MeasurableSpace Γ] [MeasurableSpace W] |
| {d : ℕ} |
|
|
| lemma norm_sub_sq_three (x y z : Point d) : ‖x-z‖^2≤2*‖x-y‖^2+2*‖y-z‖^2 := by |
| have h := norm_sub_le_norm_sub_add_norm_sub x y z |
| nlinarith [sq_nonneg (‖x-y‖-‖y-z‖),norm_nonneg (x-z),norm_nonneg (x-y),norm_nonneg (y-z)] |
|
|
| theorem SquaredAt.trans [StandardBorelSpace W] [Nonempty W] |
| {μ : Measure Ω} {ν : Measure Γ} {θ : Measure W} {f : Ω → Point d} |
| {g : Γ → Point d} {h : W → Point d} (hf : Measurable f) (hg : Measurable g) (hh : Measurable h) |
| {B C : ℝ} (h₁ : SquaredAt μ ν f g B) (h₂ : SquaredAt ν θ g h C) : |
| SquaredAt μ θ f h (2*B+2*C) := by |
| obtain ⟨κ,hκ,hl,hr,hi,he⟩ := h₁ |
| obtain ⟨ι,hι,hll,hrr,hii,hee⟩ := h₂ |
| let := hκ |
| let := hι |
| have hlink : κ.map Prod.snd=ι.fst := hr.trans hll.symm |
| let Λ := lift κ ι Prod.snd measurable_snd |
| have hp : Λ.fst=κ := lift_fst κ ι Prod.snd measurable_snd |
| have hm : Λ.map (fun z => (z.1.2,z.2))=ι := lift_map κ ι Prod.snd measurable_snd hlink |
| have hi₁ : Integrable (fun z : (Ω × Γ) × W => ‖f z.1.1-g z.1.2‖^2) Λ := by |
| have hk : Integrable (fun z : Ω × Γ => ‖f z.1-g z.2‖^2) (Λ.map Prod.fst) := by |
| change Integrable _ Λ.fst |
| rw [hp] |
| exact hi |
| exact hk.comp_measurable measurable_fst |
| have E₁ : (∫z,‖f z.1.1-g z.1.2‖^2 ∂Λ)≤B := by |
| have hm₁ : Measurable (fun z : Ω × Γ => ‖f z.1-g z.2‖^2) := by fun_prop |
| rw [←integral_map measurable_fst.aemeasurable hm₁.aestronglyMeasurable] |
| change (∫z,‖f z.1-g z.2‖^2 ∂Λ.fst)≤B |
| rw [hp] |
| exact he |
| have H₂ := lift_integrable κ ι Prod.snd measurable_snd hlink _ hii |
| have hi₂ : Integrable (fun z : (Ω × Γ) × W => ‖g z.1.2-h z.2‖^2) Λ := H₂.1 |
| have E₂ : (∫z,‖g z.1.2-h z.2‖^2 ∂Λ)≤C := H₂.2.trans_le hee |
| have hdom := (hi₁.const_mul 2).add (hi₂.const_mul 2) |
| have hb (z : (Ω × Γ) × W) : ‖f z.1.1-h z.2‖^2≤ |
| 2*‖f z.1.1-g z.1.2‖^2+2*‖g z.1.2-h z.2‖^2 := norm_sub_sq_three _ _ _ |
| have hcost : Measurable (fun z : (Ω × Γ) × W => ‖f z.1.1-h z.2‖^2) := by fun_prop |
| have hnew := hdom.mono' hcost.aestronglyMeasurable |
| (Filter.Eventually.of_forall fun z => by simpa only [Real.norm_eq_abs,abs_sq,Pi.add_apply] using hb z) |
| refine SquaredAt.of_joint Λ μ θ (fun z => z.1.1) Prod.snd (by fun_prop) measurable_snd |
| ?_ ?_ f h hf hh hnew ?_ |
| · change Measure.map (Prod.fst ∘ Prod.fst) Λ=μ |
| rw [←Measure.map_map measurable_fst measurable_fst] |
| change Λ.fst.map Prod.fst=μ |
| rw [hp] |
| exact hl |
| · have hs := congrArg (Measure.map Prod.snd) hm |
| rw [Measure.map_map measurable_snd (by fun_prop)] at hs |
| exact hs.trans hrr |
| · apply (integral_mono hnew hdom hb).trans |
| simp only [Pi.add_apply] |
| rw [integral_add (hi₁.const_mul 2) (hi₂.const_mul 2),integral_const_mul,integral_const_mul] |
| linarith |
| |
| lemma SquaredAt.same_tape (μ : Measure Ω) [IsProbabilityMeasure μ] |
| (f g : Ω → Point d) (hf : Measurable f) (hg : Measurable g) {B : ℝ} |
| (hi : Integrable (fun z => ‖f z-g z‖^2) μ) (he : (∫z,‖f z-g z‖^2 ∂μ)≤B) : |
| SquaredAt μ μ f g B := |
| SquaredAt.of_joint μ μ μ id id measurable_id measurable_id (Measure.map_id) (Measure.map_id) |
| f g hf hg hi he |
| end LogConcaveSampling.Coupling |
| |
| end |
| |
| end |
| |
| end OAI |
| |
