| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.CompositeAngle |
| import OAI.Probability.LogConcave.Complexity.HarmonicCorrectionBudget |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
| open TensorEnergy Quadrature |
|
|
| local instance : DecidableEq Unit := Classical.decEq _ |
|
|
| def angleNodes (n : ℕ) (i : Fin (n+1)) : ℝ := (i:ℝ) |
|
|
| lemma angleNodes_injective (n : ℕ) : Function.Injective (angleNodes n) := by |
| intro i j h |
| apply Fin.ext |
| change (i:ℝ)=(j:ℝ) at h |
| exact_mod_cast h |
|
|
| lemma liePathJet_measurable {S : Type} [Fintype S] {d : ℕ} {H : Point d → ℝ} |
| (hH : PolySmooth H) (A : (Unit ⊕ Unit → Fin d) → Point d → ℝ) |
| (F : (S → Fin d) → Point d → ℝ) (hA : ∀c,PolySmooth (A c)) (hF : ∀c,PolySmooth (F c)) |
| (Ξ : Point d → ℝ → Point d) (hm : Measurable (fun p : ℝ × Point d => Ξ p.2 p.1)) |
| (k : ℕ) (v : ℝ) : |
| Measurable (fun y => tensorVector (iterTensorLie H A F k) (Ξ y v)) := |
| (tensorVector_contDiff _ (fun c => (iterTensorLie_polySmooth hH A F hA hF k c).smooth)).continuous.measurable.comp |
| (hm.comp (measurable_const.prodMk measurable_id)) |
|
|
| lemma chainTaylor_measurable {Ω E : Type*} [MeasurableSpace Ω] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [SecondCountableTopology E] (J : ℕ → ℝ → Ω → E) |
| (hm : ∀k t,Measurable (J k t)) (n : ℕ) (a b : ℝ) : |
| Measurable (fun y => chainTaylor (fun k t => J k t y) n a b) := by |
| unfold chainTaylor |
| exact Finset.measurable_sum _ (fun k _ => (hm k a).const_smul _) |
|
|
| theorem harmonic_correction_stencil_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T : ℝ} |
| (hr : 0<r) (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) |
| (hR : 0<R) (hRT : R^2≤1-T^2) (hT0 : 0≤T) (hT1 : T<1) |
| (s t u : Icc (0:ℝ) T) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) |
| (hder : ∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r u) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v) |
| (hm : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1)) |
| (hlaw : ∀v∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r u)).map (fun y => Ξ y v)= |
| gibbs (centeringPotential F x r u)) : |
| let A := jointCorrectionArray (fun z => probabilityTransport hF x hr.le hl hT0 hT1 s t z-z) |
| let J := fun k v y => tensorVector (iterTensorLie (centeringPotential F x r u) (harmonicSkew d) A k) (Ξ y v) |
| ∀n : ℕ,1≤n → ∀ψ : ℝ,0<ψ → ψ*n≤1 → |
| Integrable (fun y => ‖derivativeStencil (angleNodes n) ψ (fun i => J 0 (ψ*angleNodes n i) y)-J 1 0 y‖^2) |
| (gibbs (centeringPotential F x r u)) ∧ |
| (∫y,‖derivativeStencil (angleNodes n) ψ (fun i => J 0 (ψ*angleNodes n i) y)-J 1 0 y‖^2 |
| ∂gibbs (centeringPotential F x r u))≤ |
| ((∑i,|derivativeWeight (angleNodes n) i|)/ψ)^2* |
| (((ψ*n)^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r^2)^2)*(R⁻¹)^(4*(n+1))*harmonicCorrectionBudget (n+1))) := by |
| intro A J n hn ψ hψ hψn |
| have hu1 := u.2.2.trans_lt hT1 |
| have hH : PolySmooth (centeringPotential F x r u) := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl u.2.1 hu1) (gaussianPotential_polySmooth d) |
| have hJ (k : ℕ) (v : ℝ) : Measurable (J k v) := |
| liePathJet_measurable hH (harmonicSkew d) A (harmonicSkew_polySmooth d) |
| (jointCorrectionArray_polySmooth hF x hr hlam hl hR hRT hT0 hT1 s t) Ξ hm k v |
| have hnode (i : Fin (n+1)) : 0≤ψ*angleNodes n i ∧ ψ*angleNodes n i≤ψ*n := by |
| constructor |
| · exact mul_nonneg hψ.le (Nat.cast_nonneg _) |
| · apply mul_le_mul_of_nonneg_left _ hψ.le |
| change (i:ℝ)≤(n:ℝ) |
| exact_mod_cast Nat.le_of_lt_succ i.2 |
| have htay (i : Fin (n+1)) := harmonic_correction_taylor_rms hF x hr hlam hl hR hRT hT0 hT1 |
| s t u Ξ hder hm hlaw n 0 (ψ*angleNodes n i) le_rfl (hnode i).1 ((hnode i).2.trans hψn) |
| apply derivative_stencil_error_rms (angleNodes n) (angleNodes_injective n) n hn (by simp) hψ J |
| · intro i |
| exact ((hJ 0 _).sub (chainTaylor_measurable J hJ n 0 _)).aestronglyMeasurable |
| · intro i |
| simpa only [J,iterTensorLie,sub_zero] using (htay i).1 |
| · intro i |
| have hh := (htay i).2 |
| change (∫y,‖J 0 (ψ*angleNodes n i) y-chainTaylor |
| (fun k v => J k v y) n 0 (ψ*angleNodes n i)‖^2 ∂gibbs (centeringPotential F x r u))≤_ at hh |
| apply hh.trans |
| simp only [sub_zero] |
| apply mul_le_mul_of_nonneg_right _ (by positivity [harmonicCorrectionBudget_nonneg (n+1)]) |
| apply pow_le_pow_left₀ (div_nonneg (pow_nonneg (hnode i).1 _) (Nat.cast_nonneg _)) |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) |
| exact pow_le_pow_left₀ (hnode i).1 (hnode i).2 _ |
| 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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