| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.JetMaterial |
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|
| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
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|
| lemma joint_mean_material {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0 < r) |
| (hlam : 0 < lam) (hl : (lam:ℝ)*r^2≤1/2) (h0 : 0≤ρ) (h1 : ρ<1) |
| (i : Fin d) (y : Point d) : |
| JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointMean F x r i) (ρ,y)= |
| JetCalculus.gadj jointSpace (jointScore F x r) (fun k => jointU F x r ((lam:ℝ)*r) |
| (fun _ : Unit => JetCalculus.spaceBasis d k) [()] i) (ρ,y) := by |
| have hm := jointMean_timeSmooth hF x hr.le hl i (ρ,y) (by dsimp; linarith) h1 |
| have ht := joint_mdir_eq_material F x r (hm.differentiableAt (by simp)) |
| change JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointMean F x r i) (ρ,y)=_ at ht |
| rw [ht] |
| change materialScalar F x r (fun t y => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) |
| (conditionalFieldMean F x r t y)) ρ y=_ |
| rw [material_conditionalFieldMean hF x hr hlam hl h0 h1] |
| rw [joint_gadj_slice hF x hr.le hl h0 h1 (fun k => |
| (jointU_timeSmooth hF x hr hlam hl (fun _ : Unit => JetCalculus.spaceBasis d k) [()] |
| (by simp) i (ρ,y) (by dsimp; linarith) h1).differentiableAt (by simp))] |
| rfl |
|
|
| lemma joint_mean_outer {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0 < r) |
| (hlam : 0 < lam) (hl : (lam:ℝ)*r^2≤1/2) (h0 : 0≤ρ) (h1 : ρ<1) |
| (i : Fin d) (y : Point d) : |
| JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointMean F x r i) (ρ,y)= |
| ((lam:ℝ)*r)*(TensorSum.pure TensorAtom.one).outer F x r ((lam:ℝ)*r) (fun _ => i) (ρ,y) := by |
| rw [joint_mean_material hF x hr hlam hl h0 h1] |
| have hL : (lam:ℝ)*r≠0 := ne_of_gt (mul_pos (by exact_mod_cast hlam) hr) |
| have he : (fun k => jointU F x r ((lam:ℝ)*r) |
| (fun _ : Unit => JetCalculus.spaceBasis d k) [()] i)= |
| fun k p => ((lam:ℝ)*r)*(TensorSum.pure TensorAtom.one).eval F x r ((lam:ℝ)*r) |
| (Sum.elim (fun _ => i) (fun _ => k)) p := by |
| funext k p |
| rw [TensorSum.pure_eval,TensorAtom.one_eval] |
| dsimp only [Sum.elim_inl,Sum.elim_inr] |
| field_simp |
| rw [he,JetCalculus.gadj_scalar_at _ _ (fun k => |
| ((TensorSum.pure TensorAtom.one).timeSmooth hF x hr hlam hl _ (ρ,y) |
| (by dsimp; linarith) h1).differentiableAt (by simp))] |
| rfl |
|
|
| theorem mean_material_representation (n : ℕ) : |
| ∃A : TensorSum (Unit ⊕ Unit), A.weightLE (2*n) ∧ |
| ∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, Primitive F lam → |
| ∀(x : Point d) {r ρ : ℝ},0 < r → 0 < lam → (lam:ℝ)*r^2≤1/2 → 0≤ρ → ρ<1 → |
| ∀(i : Fin d) (y : Point d), |
| JetCalculus.materialIter (1,0) jointSpace r (jointMean F x r) (n+1) (jointMean F x r i) (ρ,y)= |
| ((lam:ℝ)*r)*A.outer F x r ((lam:ℝ)*r) (fun _ => i) (ρ,y) := by |
| induction n with |
| | zero => |
| refine ⟨TensorSum.pure TensorAtom.one,TensorSum.weightLE_pure _ (by rfl),?_⟩ |
| intro d F lam hF x r ρ hr hlam hl h0 h1 i y |
| exact joint_mean_outer hF x hr hlam hl h0 h1 i y |
| | succ n ih => |
| obtain ⟨A,hA,he⟩ := ih |
| obtain ⟨B,hB,hAB⟩ := A.hasMaterial hA |
| refine ⟨B.add (A.insertOne.scale 1 Polynomial.X),?_,?_⟩ |
| · apply TensorSum.weightLE_add |
| · convert hB using 1 |
| · exact TensorSum.weightLE_mono (TensorSum.weightLE_scale |
| (TensorSum.weightLE_insertOne hA) _ _) (by omega) |
| · intro d F lam hF x r ρ hr hlam hl h0 h1 i y |
| have ho := A.outer_timeSmooth hF x hr hlam hl (fun _ : Unit => i) |
| have hs : JetCalculus.TimeSmooth (fun p => ((lam:ℝ)*r)*A.outer F x r ((lam:ℝ)*r) (fun _ => i) p) := |
| fun p hp0 hp1 => contDiffAt_const.mul (ho p hp0 hp1) |
| change JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) |
| (JetCalculus.materialIter (1,0) jointSpace r (jointMean F x r) (n+1) (jointMean F x r i)) (ρ,y)=_ |
| rw [JetCalculus.mdir_congr_nonneg_time (1,0) jointSpace r (jointMean F x r) |
| (JetCalculus.materialIter_timeSmooth _ _ _ (jointMean_timeSmooth hF x hr.le hl) |
| (jointMean_timeSmooth hF x hr.le hl i) (n+1)) hs |
| (fun ht0 ht1 z => he hF x hr hlam hl ht0 ht1 i z) h0 h1 y, |
| JetCalculus.mdir_const_mul_at _ _ _ _ ((ho (ρ,y) (by dsimp; linarith) h1).differentiableAt (by simp)), |
| TensorSum.outer_material hF x hr hlam hl h0 h1 hAB] |
| end LogConcaveSampling |
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| end |
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| end |
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| end |
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| end OAI |
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