| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.KernelTransportAction |
| import OAI.Probability.LogConcave.Numerics.KernelCorrectionPicard |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory RMSIntegral Quadrature |
| open scoped Classical BigOperators NNReal |
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| def kernelValueRmsBudget {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (lam Ap Ah Lp : ℝ≥0) (r R ρ h z C J : ℝ) (k n N : ℕ) : ℝ := |
| 2*(2*(Lp*probabilityMeanLipschitz lam r):ℝ≥0)^2* |
| (32*(probabilityRmsBudget F x lam Ap r h C k n N)^2+ |
| 2*(harmonicRmsBudget F x lam Ah r R ρ z J n N)^2)+ |
| 2*(probabilityRmsBudget F x lam Ap r h C k n (N+1))^2 |
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|
| theorem kernelActionPicard_numerical_rms (n : ℕ) (hn : 0<n) : |
| ∃Ap Ah Lp Lh : ℝ≥0,∃C J : ℝ,0≤C ∧ 1≤J ∧ ∃k : ℕ, |
| ∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam, |
| ∀(x : Point d) {r : ℝ},∀hr : 0<r,0<lam → ∀hl : (lam:ℝ)*r^2≤1/2,1≤d → |
| ∀{T h : ℝ},∀hT0 : 0<T,∀hT1 : T<1,∀hh : 0<h,h≤Real.log 2 → |
| Ap*probabilityMeanLipschitz lam r≤1/2 → Ah*probabilityMeanLipschitz lam r≤1/2 → |
| Lp*probabilityMeanLipschitz lam r≤1/2 → Lh*probabilityMeanLipschitz lam r≤1/2 → |
| ∀N : ℕ,∀s e : ProbabilityNode T h n, |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n e,probabilityNodeTime_mem hT0 hT1 hh hn e⟩ |
| let μ := gibbs (centeringPotential F x r S) |
| let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S E |
| ∀R : ℝ,0<R → R^2≤1-(S:ℝ)^2 → |
| ∀Ξ : Point (d+d) → ℝ → Point (d+d), |
| (∀y,Continuous (Ξ y)) → (∀y,Ξ y 0=y) → |
| (∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r S) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v) → |
| Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1) → |
| (∀v∈Icc (0:ℝ) 1,μ.map (fun y => Ξ y v)=μ) → |
| ∀m : ℕ,∀ψ : ℝ,0<ψ → ψ*m≤1 → |
| let w := fun z y => (productPointEquiv d d (Ξ y z)).1 |
| let err := fun y => kernelActionPicard F x r T h ψ n m N s e |
| (fw (productPointEquiv d d y).1,(productPointEquiv d d y).2)- |
| derivativeStencil (angleNodes m) ψ (fun i => fw (w (ψ*angleNodes m i) y)-w (ψ*angleNodes m i) y) |
| Integrable (fun y => ‖err y‖^2) μ ∧ |
| (∫y,‖err y‖^2 ∂μ)≤ |
| (∑i,|derivativeWeight (angleNodes m) i/ψ|)* |
| (∑i,|derivativeWeight (angleNodes m) i/ψ| * |
| kernelValueRmsBudget F x lam Ap Ah Lp r R S h (ψ*angleNodes m i) C J k n N) := by |
| obtain ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,hV⟩ := kernelCorrectionPicard_rms n hn |
| refine ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,?_⟩ |
| intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s e |
| dsimp only |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n e,probabilityNodeTime_mem hT0 hT1 hh hn e⟩ |
| let μ := gibbs (centeringPotential F x r S) |
| let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S E |
| intro R hR hRT Ξ hc hinit hder hm hlaw m ψ hψ hψm |
| let pos := fun y : Point (d+d) => (productPointEquiv d d y).1 |
| let mom := fun y : Point (d+d) => (productPointEquiv d d y).2 |
| let w := fun z y => pos (Ξ y z) |
| let inp := fun y => (fw (pos y),mom y) |
| let v := fun i y => kernelCorrectionPicard F x r T h (ψ*angleNodes m i) n N s e (inp y)- |
| (fw (w (ψ*angleNodes m i) y)-w (ψ*angleNodes m i) y) |
| have hz (i : Fin (m+1)) : 0≤ψ*angleNodes m i ∧ ψ*angleNodes m i≤1 := by |
| refine ⟨mul_nonneg hψ.le (Nat.cast_nonneg _),le_trans ?_ hψm⟩ |
| apply mul_le_mul_of_nonneg_left _ hψ.le |
| change (i:ℝ)≤(m:ℝ) |
| exact_mod_cast Nat.le_of_lt_succ i.2 |
| have hval (i : Fin (m+1)) : Integrable (fun y => ‖v i y‖^2) μ ∧ |
| (∫y,‖v i y‖^2 ∂μ)≤kernelValueRmsBudget F x lam Ap Ah Lp r R S h (ψ*angleNodes m i) C J k n N := |
| hV hF x hr hlam hl hd hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s e |
| R hR hRT Ξ hc hinit hder hm hlaw _ (hz i).1 (hz i).2 |
| have hpos : Measurable pos := (productPointEquiv d d).continuous.measurable.fst |
| have hmom : Measurable mom := (productPointEquiv d d).continuous.measurable.snd |
| have hfw : Measurable fw := (probabilityTransport_continuous hF x hr.le hl hT0.le hT1 S E).measurable |
| have hinp : Measurable inp := (hfw.comp hpos).prodMk hmom |
| have hw (z : ℝ) : Measurable (w z) := hpos.comp (hm.comp (measurable_const.prodMk measurable_id)) |
| have hv (i : Fin (m+1)) : Measurable (v i) := |
| ((kernelCorrectionPicard_continuous hF x hr.le hl hT0 hT1 hh _ n hn N s e).measurable.comp hinp).sub |
| ((hfw.comp (hw _)).sub (hw _)) |
| have he (y : Point (d+d)) : kernelActionPicard F x r T h ψ n m N s e (inp y)- |
| derivativeStencil (angleNodes m) ψ (fun i => fw (w (ψ*angleNodes m i) y)-w (ψ*angleNodes m i) y)= |
| ∑i,(derivativeWeight (angleNodes m) i/ψ) • v i y := by |
| simp only [kernelActionPicard,derivativeStencil,←Finset.sum_sub_distrib,←smul_sub] |
| rfl |
| have hh := weighted_sum_sq_varying (fun i => derivativeWeight (angleNodes m) i/ψ) v _ |
| (fun i => (hv i).aestronglyMeasurable) (fun i => (hval i).1) (fun i => (hval i).2) |
| simpa only [←he] using hh |
| 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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