| import Mathlib |
| import OAI.Probability.SKGap.Gaussian.StandardGaussianProduct |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set |
| open scoped NNReal ENNReal |
| open Filter |
| open scoped Topology |
| namespace SKGap |
|
|
| def coordinateDeriv {n : ℕ} (i : Fin (n+1)) (f : (Fin (n+1) → ℝ) → ℝ) |
| (x : Fin (n+1) → ℝ) : ℝ := |
| deriv (fun z => f (i.insertNth z (i.removeNth x))) (x i) |
|
|
| lemma lipschitz_insertNth {n : ℕ} (i : Fin (n+1)) (u : Fin n → ℝ) : |
| LipschitzWith 1 (fun z : ℝ => i.insertNth (α := fun _ => ℝ) z u) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| simp only [NNReal.coe_one,one_mul] |
| apply (dist_pi_le_iff dist_nonneg).2 |
| refine i.forall_iff_succAbove.mpr ⟨?_,?_⟩ |
| · simp only [Fin.insertNth_apply_same,le_refl] |
| · intro k |
| simp only [Fin.insertNth_apply_succAbove,dist_self] |
| exact dist_nonneg |
|
|
| lemma continuous_insertNth_uncurry {n : ℕ} (i : Fin (n+1)) : |
| Continuous (fun p : (Fin n → ℝ) × ℝ => i.insertNth (α := fun _ => ℝ) p.2 p.1) := by |
| apply continuous_pi |
| refine i.forall_iff_succAbove.mpr ⟨?_,?_⟩ |
| · simpa only [Fin.insertNth_apply_same] using (continuous_snd : Continuous (fun p : (Fin n → ℝ) × ℝ => p.2)) |
| · intro k |
| simpa only [Fin.insertNth_apply_succAbove,Function.comp_def] using (continuous_apply k).comp continuous_fst |
|
|
| lemma measurable_coordinateDeriv {n : ℕ} (i : Fin (n+1)) {f : (Fin (n+1) → ℝ) → ℝ} |
| (hf : Continuous f) : Measurable (coordinateDeriv i f) := by |
| have h := measurable_deriv_with_param (f := fun u z => f (i.insertNth z u)) |
| (hf.comp (continuous_insertNth_uncurry i)) |
| have hc : Continuous (fun x : Fin (n+1) → ℝ => (i.removeNth x,x i)) := by |
| apply Continuous.prodMk |
| · apply continuous_pi |
| intro k |
| exact continuous_apply (i.succAbove k) |
| · exact continuous_apply i |
| exact h.comp hc.measurable |
|
|
| lemma norm_coordinateDeriv_le {n : ℕ} (i : Fin (n+1)) {f : (Fin (n+1) → ℝ) → ℝ} |
| {L : ℝ≥0} (hf : LipschitzWith L f) (x : Fin (n+1) → ℝ) : |
| ‖coordinateDeriv i f x‖ ≤ L := by |
| have hl := hf.comp (lipschitz_insertNth i (i.removeNth x)) |
| simpa only [mul_one,coordinateDeriv,Function.comp_def] using norm_deriv_le_of_lipschitz hl (x₀ := x i) |
|
|
| theorem gaussianProduct_lipschitz_IBP {n : ℕ} (i : Fin (n+1)) |
| {f : (Fin (n+1) → ℝ) → ℝ} {L : ℝ≥0} {C : ℝ} |
| (hf : LipschitzWith L f) (hC : ∀ x, ‖f x‖ ≤ C) : |
| (∫ x, x i * f x ∂standardGaussianProduct (n+1)) = |
| ∫ x, coordinateDeriv i f x ∂standardGaussianProduct (n+1) := by |
| let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) => ℝ) i |
| have hp : MeasurePreserving e.symm |
| ((gaussianReal 0 1).prod (standardGaussianProduct n)) |
| (standardGaussianProduct (n+1)) := |
| (measurePreserving_piFinSuccAbove (fun _ => gaussianReal 0 1) i).symm |
| have hi : Integrable (fun x : Fin (n+1) → ℝ => x i * f x) |
| (standardGaussianProduct (n+1)) := by |
| simpa only [mul_comm,standardGaussianProduct] using |
| (integrable_eval (i := i) IsGaussian.integrable_id).bdd_mul |
| hf.continuous.aestronglyMeasurable (ae_of_all _ hC) |
| have hi' : Integrable (coordinateDeriv i f) (standardGaussianProduct (n+1)) := |
| Integrable.of_bound (measurable_coordinateDeriv i hf.continuous).aestronglyMeasurable (L:ℝ) |
| (ae_of_all _ (norm_coordinateDeriv_le i hf)) |
| have hsplit (g : (Fin (n+1) → ℝ) → ℝ) (hg : Integrable g (standardGaussianProduct (n+1))) : |
| (∫ x, g x ∂standardGaussianProduct (n+1)) = |
| ∫ u, ∫ z, g (i.insertNth (α := fun _ => ℝ) z u) ∂gaussianReal 0 1 ∂standardGaussianProduct n := by |
| have hi2 : Integrable (fun x => g (e.symm x)) |
| ((gaussianReal 0 1).prod (standardGaussianProduct n)) := hp.integrable_comp_of_integrable hg |
| rw [← hp.integral_comp' g,integral_prod _ hi2,integral_integral_swap hi2] |
| rfl |
| rw [hsplit _ hi,hsplit _ hi'] |
| apply integral_congr_ae |
| filter_upwards [] with u |
| simp only [coordinateDeriv,Fin.insertNth_apply_same,Fin.removeNth_insertNth] |
| simpa only [sub_zero,NNReal.coe_one,one_mul,Function.comp_def] using |
| gaussian_lipschitz_IBP (d := 0) (s := 1) (by norm_num) |
| (hf.comp (lipschitz_insertNth i u)) (fun z => hC (i.insertNth (α := fun _ => ℝ) z u)) |
| |
| variable {κ : Type*} [Fintype κ] [DecidableEq κ] |
| |
| def finiteCoordinateDeriv (i : κ) (f : (κ → ℝ) → ℝ) (x : κ → ℝ) : ℝ := |
| deriv (fun z => f (Function.update x i z)) (x i) |
| |
| lemma lipschitz_coordinate_update (i : κ) (x : κ → ℝ) : |
| LipschitzWith 1 (fun z : ℝ => Function.update x i z) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro z w |
| simp only [NNReal.coe_one,one_mul] |
| apply (dist_pi_le_iff dist_nonneg).2 |
| intro k |
| by_cases h : k = i |
| · subst k; simp |
| · simp only [Function.update_of_ne h,dist_self]; exact dist_nonneg |
| |
| lemma measurable_finiteCoordinateDeriv (i : κ) {f : (κ → ℝ) → ℝ} |
| (hf : Continuous f) : Measurable (finiteCoordinateDeriv i f) := by |
| have h := measurable_deriv_with_param (f := fun x z => f (Function.update x i z)) |
| (hf.comp (continuous_update i)) |
| exact h.comp ((continuous_id.prodMk (continuous_apply i)).measurable) |
| |
| lemma norm_finiteCoordinateDeriv_le (i : κ) {f : (κ → ℝ) → ℝ} |
| {L : ℝ≥0} (hf : LipschitzWith L f) (x : κ → ℝ) : |
| ‖finiteCoordinateDeriv i f x‖ ≤ L := by |
| simpa only [finiteCoordinateDeriv,Function.comp_def,mul_one] using |
| norm_deriv_le_of_lipschitz (hf.comp (lipschitz_coordinate_update i x)) (x₀ := x i) |
| |
| lemma gaussianFinite_lipschitz_IBP_equiv {n : ℕ} (e : Fin (n+1) ≃ κ) (i : κ) |
| {f : (κ → ℝ) → ℝ} {L : ℝ≥0} {C : ℝ} |
| (hf : LipschitzWith L f) (hC : ∀ x, ‖f x‖ ≤ C) : |
| (∫ x, x i*f x ∂Measure.pi (fun _ : κ => gaussianReal 0 1)) = |
| ∫ x, finiteCoordinateDeriv i f x ∂Measure.pi (fun _ : κ => gaussianReal 0 1) := by |
| let E := MeasurableEquiv.piCongrLeft (fun _ : κ => ℝ) e |
| have he (x : Fin (n+1) → ℝ) (k : κ) : E x k = x (e.symm k) := by |
| simp [E,MeasurableEquiv.coe_piCongrLeft,Equiv.piCongrLeft_apply] |
| have hl : LipschitzWith 1 E := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| simp only [NNReal.coe_one,one_mul] |
| apply (dist_pi_le_iff dist_nonneg).2 |
| intro k |
| simpa only [he] using dist_le_pi_dist x y (e.symm k) |
| have hp := measurePreserving_piCongrLeft (fun _ : κ => gaussianReal 0 1) e |
| have hh := gaussianProduct_lipschitz_IBP (e.symm i) |
| (by simpa only [mul_one,Function.comp_def] using hf.comp hl) (fun x => hC (E x)) |
| have hu (x : Fin (n+1) → ℝ) (z : ℝ) : |
| E ((e.symm i).insertNth z ((e.symm i).removeNth x)) = |
| Function.update (E x) i z := by |
| rw [Fin.insertNth_removeNth] |
| ext k |
| simp only [he] |
| by_cases h : k = i |
| · subst k; simp |
| · simp [he,Function.update_of_ne h,show e.symm k ≠ e.symm i from fun h' => h (e.symm.injective h')] |
| have hd (x : Fin (n+1) → ℝ) : |
| coordinateDeriv (e.symm i) (f ∘ E) x = finiteCoordinateDeriv i f (E x) := by |
| simp only [coordinateDeriv,finiteCoordinateDeriv,Function.comp_def,hu,he] |
| rw [← hp.integral_comp' (fun x => x i*f x), |
| ← hp.integral_comp' (finiteCoordinateDeriv i f)] |
| change (∫ x, E x i*f (E x) ∂standardGaussianProduct (n+1)) = |
| ∫ x, finiteCoordinateDeriv i f (E x) ∂standardGaussianProduct (n+1) |
| simpa only [← hd,he,Function.comp_def] using hh |
| |
| theorem gaussianFinite_lipschitz_IBP (i : κ) |
| {f : (κ → ℝ) → ℝ} {L : ℝ≥0} {C : ℝ} |
| (hf : LipschitzWith L f) (hC : ∀ x, ‖f x‖ ≤ C) : |
| (∫ x, x i*f x ∂Measure.pi (fun _ : κ => gaussianReal 0 1)) = |
| ∫ x, finiteCoordinateDeriv i f x ∂Measure.pi (fun _ : κ => gaussianReal 0 1) := by |
| have hn : Fintype.card κ ≠ 0 := by |
| let : Nonempty κ := ⟨i⟩ |
| exact Fintype.card_ne_zero |
| obtain ⟨n,hn⟩ := Nat.exists_eq_succ_of_ne_zero hn |
| let e : κ ≃ Fin (n+1) := (Fintype.equivFin κ).trans (Equiv.cast (congrArg Fin hn)) |
| exact gaussianFinite_lipschitz_IBP_equiv e.symm i hf hC |
| |
| end SKGap |
| |
| end |
| |
| end OAI |
| |
