| 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 e : ℕ} |
|
|
| theorem SquaredAt.shared_lipschitz {μ : Measure Ω} {ν : Measure Γ} |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (η : Measure W) [IsProbabilityMeasure η] |
| {f : Ω → Point d} {g : Γ → Point d} (hf : Measurable f) (hg : Measurable g) |
| {B K : ℝ} (hK : 0≤K) (hc : SquaredAt μ ν f g B) |
| (H : Point d → W → Point e) (hH : Measurable (uncurry H)) |
| (hLip : ∀x y w,‖H x w-H y w‖≤K*‖x-y‖) : |
| SquaredAt (μ.prod η) (ν.prod η) (fun z => H (f z.1) z.2) |
| (fun z => H (g z.1) z.2) (K^2*B) := by |
| obtain ⟨κ,hκ,hl,hr,hi,he⟩ := hc |
| let := hκ |
| let Λ := κ.prod η |
| have hΛf : Λ.map (fun z => (z.1.1,z.2))=μ.prod η := by |
| rw [show (fun z : (Ω × Γ) × W => (z.1.1,z.2))=Prod.map Prod.fst id from rfl] |
| rw [←Measure.map_prod_map _ _ measurable_fst measurable_id,Measure.map_id] |
| exact congrArg (fun m => m.prod η) hl |
| have hΛg : Λ.map (fun z => (z.1.2,z.2))=ν.prod η := by |
| rw [show (fun z : (Ω × Γ) × W => (z.1.2,z.2))=Prod.map Prod.snd id from rfl] |
| rw [←Measure.map_prod_map _ _ measurable_snd measurable_id,Measure.map_id] |
| exact congrArg (fun m => m.prod η) hr |
| have hm : Measurable (fun z : Ω × Γ => ‖f z.1-g z.2‖^2) := by fun_prop |
| have hi' : Integrable (fun z : (Ω × Γ) × W => ‖f z.1.1-g z.1.2‖^2) Λ := hi.comp_fst η |
| have hp (z : (Ω × Γ) × W) : ‖H (f z.1.1) z.2-H (g z.1.2) z.2‖^2≤ |
| K^2*‖f z.1.1-g z.1.2‖^2 := by |
| simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _) (hLip _ _ _) 2 |
| have hcmeas : Measurable (fun z : (Ω × Γ) × W => ‖H (f z.1.1) z.2-H (g z.1.2) z.2‖^2) := by |
| have h₁ : Measurable (fun z : (Ω × Γ) × W => H (f z.1.1) z.2) := hH.comp ((hf.comp (measurable_fst.comp measurable_fst)).prodMk measurable_snd) |
| have h₂ : Measurable (fun z : (Ω × Γ) × W => H (g z.1.2) z.2) := hH.comp ((hg.comp (measurable_snd.comp measurable_fst)).prodMk measurable_snd) |
| exact (h₁.sub h₂).norm.pow_const 2 |
| have hdom := hi'.const_mul (K^2) |
| have hIi := hdom.mono' hcmeas.aestronglyMeasurable (Filter.Eventually.of_forall fun z => by |
| simpa only [Real.norm_eq_abs,abs_sq] using hp z) |
| refine SquaredAt.of_joint Λ (μ.prod η) (ν.prod η) |
| (fun z => (z.1.1,z.2)) (fun z => (z.1.2,z.2)) (by fun_prop) (by fun_prop) |
| hΛf hΛg _ _ ?_ ?_ hIi ?_ |
| · exact hH.comp ((hf.comp measurable_fst).prodMk measurable_snd) |
| · exact hH.comp ((hg.comp measurable_fst).prodMk measurable_snd) |
| · apply (integral_mono hIi hdom hp).trans |
| rw [integral_const_mul] |
| have hfst : ∫z : (Ω × Γ) × W, ‖f z.1.1-g z.1.2‖^2 ∂Λ = ∫z, ‖f z.1-g z.2‖^2 ∂κ := by |
| exact (measurePreserving_fst (μ:=κ) (ν:=η)).hasLaw.integral_comp hm.aestronglyMeasurable |
| rw [hfst] |
| exact mul_le_mul_of_nonneg_left he (pow_nonneg hK 2) |
|
|
| lemma SquaredAt.add_shared {μ : Measure Ω} {ν : Measure Γ} |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (η : Measure W) [IsProbabilityMeasure η] |
| {f : Ω → Point d} {g : Γ → Point d} (hf : Measurable f) (hg : Measurable g) |
| {B : ℝ} (hc : SquaredAt μ ν f g B) (h : W → Point d) (hh : Measurable h) : |
| SquaredAt (μ.prod η) (ν.prod η) (fun z => f z.1+h z.2) (fun z => g z.1+h z.2) B := by |
| simpa only [one_pow,one_mul] using hc.shared_lipschitz η hf hg (by norm_num : (0:ℝ)≤1) |
| (fun x w => x+h w) (by fun_prop) (fun x y w => by rw [add_sub_add_right_eq_sub,one_mul]) |
|
|
| end LogConcaveSampling.Coupling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical |
|
|
| variable {X Ω Γ : Type*} [MeasurableSpace X] [MeasurableSpace Ω] [MeasurableSpace Γ] |
| {d : ℕ} |
|
|
| theorem SquaredAt.mixture (μ : Measure X) [IsProbabilityMeasure μ] |
| (ν : Measure Ω) [IsProbabilityMeasure ν] (θ : Measure Γ) [IsProbabilityMeasure θ] |
| (K : Kernel X (Ω × Γ)) [IsMarkovKernel K] |
| (hl : ∀x,(K x).fst=ν) (hr : ∀x,(K x).snd=θ) |
| (f : X → Ω → Point d) (g : X → Γ → Point d) |
| (hf : Measurable (uncurry f)) (hg : Measurable (uncurry g)) |
| (B : X → ℝ) (hB : Integrable B μ) |
| (hi : ∀x,Integrable (fun z => ‖f x z.1-g x z.2‖^2) (K x)) |
| (he : ∀x,(∫z,‖f x z.1-g x z.2‖^2 ∂K x)≤B x) : |
| SquaredAt (μ.prod ν) (μ.prod θ) (uncurry f) (uncurry g) (∫x,B x ∂μ) := by |
| let Λ := μ ⊗ₘ K |
| let F : X × (Ω × Γ) → ℝ := fun z => ‖f z.1 z.2.1-g z.1 z.2.2‖^2 |
| have hF : Measurable F := by unfold F; fun_prop |
| have hFi : Integrable (fun x => ∫z,F (x,z) ∂K x) μ := by |
| apply hB.mono' hF.stronglyMeasurable.integral_kernel_prod_right'.aestronglyMeasurable |
| exact Filter.Eventually.of_forall (fun x => by |
| rw [Real.norm_of_nonneg (integral_nonneg fun _ => sq_nonneg _)] |
| exact he x) |
| have hI : Integrable F Λ := by |
| apply (Measure.integrable_compProd_iff hF.aestronglyMeasurable).mpr |
| refine ⟨Filter.Eventually.of_forall hi,?_⟩ |
| simpa only [F,Real.norm_eq_abs,abs_sq] using hFi |
| have hleft : K.map Prod.fst=Kernel.const X ν := by |
| ext x s hs |
| rw [Kernel.map_apply _ measurable_fst] |
| exact congrArg (fun m : Measure Ω => m s) (hl x) |
| have hright : K.map Prod.snd=Kernel.const X θ := by |
| ext x s hs |
| rw [Kernel.map_apply _ measurable_snd] |
| exact congrArg (fun m : Measure Γ => m s) (hr x) |
| apply SquaredAt.of_joint Λ (μ.prod ν) (μ.prod θ) |
| (fun z => (z.1,z.2.1)) (fun z => (z.1,z.2.2)) (by fun_prop) (by fun_prop) |
| ?_ ?_ (uncurry f) (uncurry g) hf hg hI |
| · change (∫z,F z ∂Λ)≤_ |
| rw [Measure.integral_compProd hI] |
| exact integral_mono hFi hB he |
| · change Λ.map (Prod.map id Prod.fst)=_ |
| dsimp only [Λ] |
| rw [←Measure.compProd_map measurable_fst,hleft,Measure.compProd_const] |
| · change Λ.map (Prod.map id Prod.snd)=_ |
| dsimp only [Λ] |
| rw [←Measure.compProd_map measurable_snd,hright,Measure.compProd_const] |
| end LogConcaveSampling.Coupling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Function Set TopologicalSpace |
| open scoped Classical |
|
|
| variable {X Ω Γ : Type*} [PseudoMetricSpace X] [SeparableSpace X] [Nonempty X] |
| [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Ω] [MeasurableSpace Γ] |
| {d : ℕ} |
|
|
| theorem SquaredAt.random_anchor (μ : Measure X) [IsProbabilityMeasure μ] |
| (ν : Measure Ω) [IsProbabilityMeasure ν] (θ : Measure Γ) [IsProbabilityMeasure θ] |
| (f : X → Ω → Point d) (g : X → Γ → Point d) |
| (hf : Measurable (uncurry f)) (hg : Measurable (uncurry g)) |
| (ρ : X → ℝ) (hρ : ∀x,1≤ρ x) (hρlip : ∀x y,ρ y≤ρ x+dist x y) |
| (hρi : Integrable (fun x => (ρ x)^2) μ) |
| {Lf Lg E : ℝ} (hLf : 0≤Lf) (hLg : 0≤Lg) (hE : 0<E) |
| (hfl : ∀x y z,‖f x z-f y z‖≤Lf*dist x y) |
| (hgl : ∀x y z,‖g x z-g y z‖≤Lg*dist x y) |
| (hc : ∀x,SquaredAt ν θ (f x) (g x) (E^2*(ρ x)^2)) : |
| SquaredAt (μ.prod ν) (μ.prod θ) (uncurry f) (uncurry g) |
| (10*E^2*(∫x,(ρ x)^2 ∂μ)) := by |
| let e := min 1 (E/(Lf+Lg+1)) |
| have hden : 0<Lf+Lg+1 := by linarith |
| have he : 0<e := lt_min (by norm_num) (div_pos hE hden) |
| have he1 : e≤1 := min_le_left _ _ |
| have heE : (Lf+Lg)*e≤E := by |
| have hh := (le_div_iff₀ hden).mp (min_le_right (1:ℝ) (E/(Lf+Lg+1))) |
| change e*(Lf+Lg+1)≤E at hh |
| nlinarith |
| obtain ⟨q,hq,hqe,K,hK,hcouple⟩ := measurable_coupling_of_pointwise ν θ f g hf hg |
| (fun x => E^2*(ρ x)^2) hLf hLg he hfl hgl hc |
| let := hK |
| have hb (x) : 2*(E^2*(ρ (denseSeq X (q x)))^2)+2*((Lf+Lg)*e)^2≤10*E^2*(ρ x)^2 := by |
| have hh : ρ (denseSeq X (q x)) ≤ ρ x + e := by |
| calc |
| _ ≤ ρ x + dist x (denseSeq X (q x)) := hρlip _ _ |
| _ ≤ ρ x + e := by gcongr; exact (hqe x).le |
| have hρx := hρ x |
| have hρy := hρ (denseSeq X (q x)) |
| have hy : (ρ (denseSeq X (q x)))^2≤4*(ρ x)^2 := by nlinarith |
| have hys := mul_le_mul_of_nonneg_left hy (sq_nonneg E) |
| have hz : ((Lf+Lg)*e)^2≤E^2 := pow_le_pow_left₀ (by positivity) heE 2 |
| nlinarith [sq_nonneg E,mul_nonneg (sq_nonneg E) (show 0≤(ρ x)^2-1 by nlinarith)] |
| have hm := SquaredAt.mixture μ ν θ K (fun x => (hcouple x).1) (fun x => (hcouple x).2.1) |
| f g hf hg (fun x => 10*E^2*(ρ x)^2) (hρi.const_mul _) (fun x => (hcouple x).2.2.1) |
| (fun x => (hcouple x).2.2.2.trans (hb x)) |
| simpa only [integral_const_mul] using hm |
|
|
| end LogConcaveSampling.Coupling |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
