| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.TerminalMean |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory TopologicalSpace Set |
| open scoped Classical |
|
|
| variable {X Ω Γ : Type*} [PseudoMetricSpace X] [SeparableSpace X] [Nonempty X] |
| [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Ω] [MeasurableSpace Γ] |
|
|
| theorem exists_countable_quantizer {e : ℝ} (he : 0<e) : |
| ∃q : X → ℕ,Measurable q ∧ ∀x,dist x (denseSeq X (q x))<e := by |
| have hp : ∀x : X, ∃n,dist x (denseSeq X n)<e := |
| fun x => (denseRange_denseSeq X).exists_dist_lt x he |
| refine ⟨fun x => Nat.find (hp x),measurable_find hp ?_,fun x => Nat.find_spec (hp x)⟩ |
| intro n |
| exact (continuous_id.dist continuous_const).measurable measurableSet_Iio |
|
|
| def SquaredAt {d : ℕ} (μ : Measure Ω) (ν : Measure Γ) |
| (f : Ω → Point d) (g : Γ → Point d) (B : ℝ) : Prop := |
| ∃κ : Measure (Ω × Γ),IsProbabilityMeasure κ ∧ κ.fst=μ ∧ κ.snd=ν ∧ |
| Integrable (fun z => ‖f z.1-g z.2‖^2) κ ∧ (∫z,‖f z.1-g z.2‖^2 ∂κ)≤B |
|
|
| theorem measurable_coupling_of_pointwise {d : ℕ} |
| (μ : Measure Ω) (ν : Measure Γ) (f : X → Ω → Point d) (g : X → Γ → Point d) |
| (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) |
| (B : X → ℝ) {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 μ ν (f x) (g x) (B x)) : |
| ∃q : X → ℕ,Measurable q ∧ (∀x,dist x (denseSeq X (q x))<e) ∧ |
| ∃K : Kernel X (Ω × Γ),IsMarkovKernel K ∧ |
| ∀x,(K x).fst=μ ∧ (K x).snd=ν ∧ |
| Integrable (fun z => ‖f x z.1-g x z.2‖^2) (K x) ∧ |
| (∫z,‖f x z.1-g x z.2‖^2 ∂K x)≤2*B (denseSeq X (q x))+2*((Lf+Lg)*e)^2 := by |
| obtain ⟨q,hq,hqe⟩ := exists_countable_quantizer (X:=X) he |
| choose κ hκ hfst hsnd hi hb using (fun n => hc (denseSeq X n)) |
| let K : Kernel X (Ω × Γ) := ⟨fun x => κ (q x),(measurable_of_countable κ).comp hq⟩ |
| have : IsMarkovKernel K := ⟨fun x => hκ (q x)⟩ |
| refine ⟨q,hq,hqe,K,inferInstance,?_⟩ |
| intro x |
| have : IsProbabilityMeasure (K x) := hκ (q x) |
| let y := denseSeq X (q x) |
| let C := ((Lf+Lg)*e)^2 |
| have hdom : Integrable (fun z : Ω × Γ => 2*‖f y z.1-g y z.2‖^2+2*C) (K x) := |
| ((hi (q x)).const_mul 2).add (integrable_const _) |
| have hp (z : Ω × Γ) : ‖f x z.1-g x z.2‖^2≤2*‖f y z.1-g y z.2‖^2+2*C := by |
| have ht := norm_sub_le_norm_sub_add_norm_sub (f x z.1) (f y z.1) (g x z.2) |
| have hh := norm_sub_le_norm_sub_add_norm_sub (f y z.1) (g y z.2) (g x z.2) |
| rw [norm_sub_rev (g y z.2)] at hh |
| have hf' := (hfl x y z.1).trans (mul_le_mul_of_nonneg_left (hqe x).le hLf) |
| have hg' := (hgl x y z.2).trans (mul_le_mul_of_nonneg_left (hqe x).le hLg) |
| have hn : ‖f x z.1-g x z.2‖≤‖f y z.1-g y z.2‖+(Lf+Lg)*e := by linarith |
| have hn2 := pow_le_pow_left₀ (norm_nonneg _) hn 2 |
| dsimp only [C] |
| nlinarith [sq_nonneg (‖f y z.1-g y z.2‖-(Lf+Lg)*e)] |
| have hm : Measurable (fun z : Ω × Γ => ‖f x z.1-g x z.2‖^2) := by |
| exact ((hf.of_uncurry_left.comp measurable_fst).sub |
| (hg.of_uncurry_left.comp measurable_snd)).norm.pow_const 2 |
| have hint := hdom.mono' hm.aestronglyMeasurable (Filter.Eventually.of_forall (fun z => by |
| simpa only [Real.norm_eq_abs,abs_sq] using hp z)) |
| refine ⟨hfst (q x),hsnd (q x),hint,(integral_mono hint hdom hp).trans ?_⟩ |
| change (∫z,2*‖f (denseSeq X (q x)) z.1-g (denseSeq X (q x)) z.2‖^2+2*C ∂κ (q x))≤_ |
| rw [integral_add ((hi (q x)).const_mul 2) (integrable_const _),integral_const_mul, |
| integral_const,probReal_univ,one_smul] |
| have hh := hb (q x) |
| change 2*(∫z,‖f y z.1-g y z.2‖^2 ∂κ (q x))+2*C≤_ |
| linarith |
| end LogConcaveSampling.Coupling |
| |
| end |
| |
| end |
| |
| section |
| |
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| |
| variable {Ω A B : Type*} [MeasurableSpace Ω] [MeasurableSpace A] [MeasurableSpace B] |
| [StandardBorelSpace B] [Nonempty B] |
| |
| def lift (ν : Measure Ω) (κ : Measure (A × B)) [IsFiniteMeasure κ] |
| (T : Ω → A) (hT : Measurable T) : Measure (Ω × B) := |
| ν ⊗ₘ κ.condKernel.comap T hT |
| |
| instance lift_probability (ν : Measure Ω) [IsProbabilityMeasure ν] |
| (κ : Measure (A × B)) [IsProbabilityMeasure κ] (T : Ω → A) (hT : Measurable T) : |
| IsProbabilityMeasure (lift ν κ T hT) := by unfold lift; infer_instance |
| |
| lemma lift_fst (ν : Measure Ω) [SFinite ν] (κ : Measure (A × B)) [IsFiniteMeasure κ] |
| (T : Ω → A) (hT : Measurable T) : (lift ν κ T hT).fst=ν := by |
| exact Measure.fst_compProd _ _ |
| |
| theorem lift_map (ν : Measure Ω) [SFinite ν] (κ : Measure (A × B)) [IsFiniteMeasure κ] |
| (T : Ω → A) (hT : Measurable T) (hlaw : ν.map T=κ.fst) : |
| (lift ν κ T hT).map (fun z => (T z.1,z.2))=κ := by |
| calc |
| _ = κ.fst ⊗ₘ κ.condKernel := by |
| ext s hs |
| have hm : Measurable (fun z : Ω × B => (T z.1,z.2)) := by fun_prop |
| rw [Measure.map_apply hm hs] |
| rw [lift,Measure.compProd_apply (hs.preimage (by fun_prop)),Measure.compProd_apply hs] |
| simp only [Kernel.comap_apply] |
| rw [←hlaw] |
| rw [lintegral_map (Kernel.measurable_kernel_prodMk_left (κ := κ.condKernel) hs) hT] |
| rfl |
| _ = κ := Measure.disintegrate κ κ.condKernel |
| |
| lemma lift_integrable (ν : Measure Ω) [SFinite ν] |
| (κ : Measure (A × B)) [IsFiniteMeasure κ] (T : Ω → A) (hT : Measurable T) |
| (hlaw : ν.map T=κ.fst) (f : A × B → ℝ) (hf : Integrable f κ) : |
| Integrable (fun z => f (T z.1,z.2)) (lift ν κ T hT) ∧ |
| (∫z,f (T z.1,z.2) ∂lift ν κ T hT)=∫z,f z ∂κ := by |
| have hm : Measurable (fun z : Ω × B => (T z.1,z.2)) := by fun_prop |
| have he := lift_map ν κ T hT hlaw |
| have hi : Integrable f ((lift ν κ T hT).map (fun z => (T z.1,z.2))) := by rw [he]; exact hf |
| exact ⟨hi.comp_measurable hm,(integral_map hm.aemeasurable hi.aestronglyMeasurable).symm.trans |
| (by rw [he])⟩ |
| end LogConcaveSampling.Coupling |
| |
| end |
| |
| end |
| |
| section |
| |
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Function SeedCompiler |
| open scoped BigOperators |
| |
| variable {Ω A E : Type*} [MeasurableSpace Ω] [MeasurableSpace A] |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
| [MeasurableSpace E] [BorelSpace E] {k : ℕ} |
| |
| theorem absorb_coupling (κ : Measure Ω) [IsProbabilityMeasure κ] |
| (μ : Measure A) [IsProbabilityMeasure μ] (ν : Measure E) |
| (anchor : Ω → A) (noise center : Ω → E) |
| (ha : Measurable anchor) (hz : Measurable noise) (hc : Measurable center) |
| (hlaw : κ.map (fun w => (anchor w,noise w))=μ.prod (stdGaussian E)) |
| (x : A → E) (hx : Measurable x) {r D K S B : ℝ} |
| (hD : 0<D) (hK : 0≤K) (v : Fin k → ℝ) (hv : (∑i,v i^2)≤D^2) |
| (M : E → (Fin k → E) → E) (hm : Measurable (uncurry M)) |
| (hshift : ShiftInvariant r v M) |
| (hLip : ∀x y g,‖M x g-M y g‖≤K*‖x-y‖) |
| (Q : A → E → E) (hQ : Measurable (uncurry Q)) |
| (hci : Integrable (fun w => ‖center w-(x (anchor w)+(r/(2*D)) • noise w)‖^2) κ) |
| (hce : (∫w,‖center w-(x (anchor w)+(r/(2*D)) • noise w)‖^2 ∂κ)≤S) |
| (π : Measure ((A × (Fin k → E)) × E)) [IsProbabilityMeasure π] |
| (hπ : π.fst=μ.prod (Measure.pi fun _ : Fin k => stdGaussian E)) |
| (hπnoise : π.map (fun z => (z.1.1,z.2))=μ.prod ν) |
| (hbi : Integrable (fun z => ‖M (x z.1.1) z.1.2-Q z.1.1 z.2‖^2) π) |
| (hbe : (∫z,‖M (x z.1.1) z.1.2-Q z.1.1 z.2‖^2 ∂π)≤B) : |
| let a := fun i => v i/(2*D) |
| let P := slotTransform (sqrtCoefficient a) a |
| ∃Λ : Measure ((Ω × (Fin k → E)) × E),IsProbabilityMeasure Λ ∧ |
| Λ.fst=κ.prod (Measure.pi fun _ : Fin k => stdGaussian E) ∧ |
| Λ.map (fun z => (anchor z.1.1,z.2))=μ.prod ν ∧ |
| Integrable (fun z => ‖M (center z.1.1) (P z.1.2)-Q (anchor z.1.1) z.2‖^2) Λ ∧ |
| (∫z,‖M (center z.1.1) (P z.1.2)-Q (anchor z.1.1) z.2‖^2 ∂Λ)≤2*K^2*S+2*B := by |
| dsimp only |
| let a := fun i => v i/(2*D) |
| let γ : Measure (Fin k → E) := Measure.pi fun _ => stdGaussian E |
| let T : Ω × (Fin k → E) → A × (Fin k → E) := fun z => |
| (anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1)) |
| have hT : Measurable T := by |
| exact (ha.comp measurable_fst).prodMk ((restored (sqrtCoefficient a) a).measurable.comp |
| (measurable_snd.prodMk (hz.comp measurable_fst))) |
| have hTlaw : (κ.prod γ).map T=π.fst := by |
| rw [hπ] |
| exact restored_from_joint_law κ μ anchor noise ha hz hlaw a |
| ((normalized_shift_bound hD v hv).trans (by norm_num)) |
| let Λ := lift (κ.prod γ) π T hT |
| have hΛ : Λ.fst=κ.prod γ := lift_fst _ _ _ _ |
| have hΛmap := lift_map (κ.prod γ) π T hT hTlaw |
| have hC₀ := hci.comp_fst γ |
| have hC : Integrable (fun z : (Ω × (Fin k → E)) × E => |
| ‖center z.1.1-(x (anchor z.1.1)+(r/(2*D)) • noise z.1.1)‖^2) Λ := by |
| have h : Integrable (fun z : Ω × (Fin k → E) => |
| ‖center z.1-(x (anchor z.1)+(r/(2*D)) • noise z.1)‖^2) Λ.fst := by |
| rw [hΛ]; exact hC₀ |
| exact h.comp_measurable measurable_fst |
| have hCeq : (∫z : (Ω × (Fin k → E)) × E, |
| ‖center z.1.1-(x (anchor z.1.1)+(r/(2*D)) • noise z.1.1)‖^2 ∂Λ)= |
| ∫w,‖center w-(x (anchor w)+(r/(2*D)) • noise w)‖^2 ∂κ := by |
| have hmf : Measurable (fun z : Ω × (Fin k → E) => |
| ‖center z.1-(x (anchor z.1)+(r/(2*D)) • noise z.1)‖^2) := by |
| exact ((hc.sub ((hx.comp ha).add (hz.const_smul _))).norm.pow_const 2).comp measurable_fst |
| rw [←integral_map measurable_fst.aemeasurable hmf.aestronglyMeasurable] |
| change (∫z : Ω × (Fin k → E),‖center z.1-(x (anchor z.1)+(r/(2*D)) • noise z.1)‖^2 ∂Λ.fst)=_ |
| rw [hΛ,integral_prod _ hC₀] |
| simp only [integral_const,probReal_univ,one_smul] |
| have hJ := lift_integrable (κ.prod γ) π T hT hTlaw |
| (fun z => ‖M (x z.1.1) z.1.2-Q z.1.1 z.2‖^2) hbi |
| have hmP : Measurable (slotTransform (sqrtCoefficient a) a : (Fin k → E) → _) := by |
| unfold slotTransform slotSum; fun_prop |
| have hmeas : Measurable (fun z : (Ω × (Fin k → E)) × E => |
| ‖M (center z.1.1) (slotTransform (sqrtCoefficient a) a z.1.2)-Q (anchor z.1.1) z.2‖^2) := by |
| exact ((hm.comp ((hc.comp (measurable_fst.comp measurable_fst)).prodMk |
| (hmP.comp (measurable_snd.comp measurable_fst)))).sub |
| (hQ.comp ((ha.comp (measurable_fst.comp measurable_fst)).prodMk measurable_snd))).norm.pow_const 2 |
| have hbound (z : (Ω × (Fin k → E)) × E) : |
| ‖M (center z.1.1) (slotTransform (sqrtCoefficient a) a z.1.2)-Q (anchor z.1.1) z.2‖^2≤ |
| 2*K^2*‖center z.1.1-(x (anchor z.1.1)+(r/(2*D)) • noise z.1.1)‖^2+ |
| 2*‖M (x (T z.1).1) (T z.1).2-Q (T z.1).1 z.2‖^2 := by |
| have he₀ := hLip (center z.1.1) (x (anchor z.1.1)+(r/(2*D)) • noise z.1.1) |
| (slotTransform (sqrtCoefficient a) a z.1.2) |
| have heq := absorption_pathwise (D:=D) v M hshift (x (anchor z.1.1)) (sqrtCoefficient a) z.1.2 (noise z.1.1) |
| rw [heq] at he₀ |
| have he := he₀ |
| have ht := norm_sub_le_norm_sub_add_norm_sub |
| (M (center z.1.1) (slotTransform (sqrtCoefficient a) a z.1.2)) |
| (M (x (T z.1).1) (T z.1).2) (Q (anchor z.1.1) z.2) |
| have hp := (sq_le_sq₀ (norm_nonneg _) |
| (add_nonneg (mul_nonneg hK (norm_nonneg _)) (norm_nonneg _))).2 |
| (ht.trans (add_le_add he (le_refl _))) |
| change _≤2*K^2*‖center z.1.1-(x (anchor z.1.1)+(r/(2*D)) • noise z.1.1)‖^2+ |
| 2*‖M (x (T z.1).1) (T z.1).2-Q (anchor z.1.1) z.2‖^2 |
| nlinarith [sq_nonneg (K*‖center z.1.1-(x (anchor z.1.1)+(r/(2*D)) • noise z.1.1)‖- |
| ‖M (x (T z.1).1) (T z.1).2-Q (anchor z.1.1) z.2‖)] |
| have hdom := (hC.const_mul (2*K^2)).add (hJ.1.const_mul 2) |
| have hi := hdom.mono' hmeas.aestronglyMeasurable (Filter.Eventually.of_forall (fun z => by |
| simpa only [Real.norm_eq_abs,abs_sq,Pi.add_apply] using hbound z)) |
| refine ⟨Λ,inferInstance,hΛ,?_,hi,(integral_mono hi hdom hbound).trans ?_⟩ |
| · rw [←hπnoise,←hΛmap,Measure.map_map (by fun_prop) (by fun_prop)] |
| rfl |
| · change (∫z : (Ω × (Fin k → E)) × E, |
| 2*K^2*‖center z.1.1-(x (anchor z.1.1)+(r/(2*D)) • noise z.1.1)‖^2+ |
| 2*‖M (x (T z.1).1) (T z.1).2-Q (T z.1).1 z.2‖^2 ∂Λ)≤_ |
| rw [integral_add (hC.const_mul _) (hJ.1.const_mul _),integral_const_mul,integral_const_mul,hCeq,hJ.2] |
| exact add_le_add (mul_le_mul_of_nonneg_left hce (by positivity)) |
| (mul_le_mul_of_nonneg_left hbe (by norm_num)) |
| end LogConcaveSampling.Coupling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Coupling |
| open MeasureTheory ProbabilityTheory Function |
| open scoped Classical BigOperators NNReal |
|
|
| variable {Ω Γ W : Type*} [MeasurableSpace Ω] [MeasurableSpace Γ] [MeasurableSpace W] |
| {d : ℕ} |
|
|
| theorem SquaredAt.of_joint (Λ : Measure W) [IsProbabilityMeasure Λ] |
| (μ : Measure Ω) (ν : Measure Γ) (T : W → Ω) (U : W → Γ) |
| (hT : Measurable T) (hU : Measurable U) (hl : Λ.map T=μ) (hr : Λ.map U=ν) |
| (f : Ω → Point d) (g : Γ → Point d) (hf : Measurable f) (hg : Measurable g) |
| {B : ℝ} (hi : Integrable (fun w => ‖f (T w)-g (U w)‖^2) Λ) |
| (he : (∫w,‖f (T w)-g (U w)‖^2 ∂Λ)≤B) : SquaredAt μ ν f g B := by |
| let J : W → Ω × Γ := fun w => (T w,U w) |
| have hJ : Measurable J := hT.prodMk hU |
| have hm : Measurable (fun z : Ω × Γ => ‖f z.1-g z.2‖^2) := by fun_prop |
| refine ⟨Λ.map J,inferInstance,?_,?_,?_,?_⟩ |
| · change (Λ.map J).map Prod.fst=μ |
| rw [Measure.map_map measurable_fst hJ] |
| exact hl |
| · change (Λ.map J).map Prod.snd=ν |
| rw [Measure.map_map measurable_snd hJ] |
| exact hr |
| · exact (integrable_map_measure hm.aestronglyMeasurable hJ.aemeasurable).mpr hi |
| · rw [integral_map hJ.aemeasurable hm.aestronglyMeasurable] |
| exact he |
|
|
| lemma SquaredAt.mono {μ : Measure Ω} {ν : Measure Γ} {f : Ω → Point d} {g : Γ → Point d} |
| {B C : ℝ} (h : SquaredAt μ ν f g B) (hBC : B≤C) : SquaredAt μ ν f g C := by |
| obtain ⟨κ,hκ,hl,hr,hi,he⟩ := h |
| exact ⟨κ,hκ,hl,hr,hi,he.trans hBC⟩ |
|
|
| lemma SquaredAt.nonneg {μ : Measure Ω} {ν : Measure Γ} {f : Ω → Point d} {g : Γ → Point d} |
| {B : ℝ} (h : SquaredAt μ ν f g B) : 0≤B := by |
| obtain ⟨κ,_,_,_,_,he⟩ := h |
| exact (integral_nonneg fun _ => sq_nonneg _).trans he |
|
|
| lemma SquaredAt.lipschitz {μ : Measure Ω} {ν : Measure Γ} |
| {f : Ω → Point d} {g : Γ → Point d} (hf : Measurable f) (hg : Measurable g) |
| {B : ℝ} (h : SquaredAt μ ν f g B) {e : ℕ} (T : Point d → Point e) |
| {K : ℝ≥0} (hT : LipschitzWith K T) : |
| SquaredAt μ ν (T ∘ f) (T ∘ g) ((K:ℝ)^2*B) := by |
| obtain ⟨κ,hκ,hl,hr,hi,he⟩ := h |
| have hp (z : Ω × Γ) : ‖T (f z.1)-T (g z.2)‖^2≤(K:ℝ)^2*‖f z.1-g z.2‖^2 := by |
| have hh := hT.dist_le_mul (f z.1) (g z.2) |
| simp only [dist_eq_norm] at hh |
| simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _) hh 2 |
| have hm : Measurable (fun z : Ω × Γ => ‖T (f z.1)-T (g z.2)‖^2) := by |
| have := hT.continuous.measurable |
| fun_prop |
| have hdom := hi.const_mul ((K:ℝ)^2) |
| have hnew := hdom.mono' hm.aestronglyMeasurable (Filter.Eventually.of_forall fun z => by |
| simpa only [Real.norm_eq_abs,abs_sq] using hp z) |
| refine ⟨κ,hκ,hl,hr,hnew,(integral_mono hnew hdom hp).trans ?_⟩ |
| rw [integral_const_mul] |
| exact mul_le_mul_of_nonneg_left he (sq_nonneg _) |
| |
| lemma SquaredAt.reindex {Ω' Γ' : Type*} [MeasurableSpace Ω'] [MeasurableSpace Γ'] |
| {μ : Measure Ω} {ν : Measure Γ} {f : Ω → Point d} {g : Γ → Point d} |
| {B : ℝ} (h : SquaredAt μ ν f g B) (T : Ω → Ω') (U : Γ → Γ') |
| (hT : Measurable T) (hU : Measurable U) |
| (f' : Ω' → Point d) (g' : Γ' → Point d) (hf' : Measurable f') (hg' : Measurable g') |
| (hfe : ∀z,f' (T z)=f z) (hge : ∀z,g' (U z)=g z) : |
| SquaredAt (μ.map T) (ν.map U) f' g' B := by |
| obtain ⟨κ,hκ,hl,hr,hi,he⟩ := h |
| have := hκ |
| refine SquaredAt.of_joint κ (μ.map T) (ν.map U) (T ∘ Prod.fst) (U ∘ Prod.snd) |
| (hT.comp measurable_fst) (hU.comp measurable_snd) ?_ ?_ f' g' hf' hg' ?_ ?_ |
| · rw [←Measure.map_map hT measurable_fst]; exact congrArg (Measure.map T) hl |
| · rw [←Measure.map_map hU measurable_snd]; exact congrArg (Measure.map U) hr |
| · simpa only [Function.comp_apply,hfe,hge] using hi |
| · simpa only [Function.comp_apply,hfe,hge] using he |
| |
| end LogConcaveSampling.Coupling |
| |
| end |
| |
| end |
| |
| end OAI |
| |
