| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.ConditionalHarmonicFlow |
| import OAI.Probability.LogConcave.Complexity.KernelValueRmsBudget |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory RMSIntegral Quadrature |
| open scoped Classical BigOperators NNReal |
|
|
| theorem kernelActionPicard_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-T^2 → |
| ∀m : ℕ,1 ≤ m → ∀ψ : ℝ,0<ψ → ψ*m≤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)-kernelTransportAction fw y |
| Integrable (fun y => ‖err y‖^2) μ ∧ |
| (∫y,‖err y‖^2 ∂μ)≤ |
| 2*((∑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))+ |
| 2*(((∑i,|derivativeWeight (angleNodes m) i|)/ψ)^2* |
| (((ψ*m)^(m+1)/(m.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r^2)^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)))) := by |
| obtain ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,hNum⟩ := kernelActionPicard_numerical_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 m hm1 ψ hψ hψm |
| have hRS : R^2≤1-(S:ℝ)^2 := by |
| have hs := pow_le_pow_left₀ S.2.1 S.2.2 2 |
| linarith |
| obtain ⟨Ξ,hc,hder,hmeas,hlaw⟩ := exists_conditional_harmonic_flow hF x hr.le hl S.2.1 (S.2.2.trans_lt hT1) |
| have hnum := hNum hF x hr hlam hl hd hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s e |
| R hR hRS Ξ (fun y => (hc y).1) (fun y => (hc y).2) hder hmeas hlaw m ψ hψ hψm |
| have hex := kernel_action_between_stencil_rms hF x hr hlam hl hR hRT hT0.le hT1 S E Ξ |
| (fun y => (hc y).2) hder hmeas hlaw m hm1 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 inp := fun y => (fw (pos y),mom y) |
| let w := fun z y => pos (Ξ y z) |
| let num := fun y => kernelActionPicard F x r T h ψ n m N s e (inp y) |
| let mid := fun y => derivativeStencil (angleNodes m) ψ |
| (fun i => fw (w (ψ*angleNodes m i) y)-w (ψ*angleNodes m i) y) |
| 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 (hmeas.comp (measurable_const.prodMk measurable_id)) |
| have hnm : Measurable num := by |
| change Measurable (fun y => ∑i,(derivativeWeight (angleNodes m) i/ψ) • |
| kernelCorrectionPicard F x r T h (ψ*angleNodes m i) n N s e (inp y)) |
| apply Finset.measurable_sum |
| intro i _ |
| have hc : Continuous (fun v : Point d => (derivativeWeight (angleNodes m) i/ψ) • v) := continuous_const_smul _ |
| exact hc.measurable.comp |
| ((kernelCorrectionPicard_continuous hF x hr.le hl hT0 hT1 hh (ψ*angleNodes m i) n hn N s e).measurable.comp hinp) |
| have hmid : Measurable mid := by |
| apply Finset.measurable_sum |
| intro i _ |
| have hc : Continuous (fun v : Point d => (derivativeWeight (angleNodes m) i/ψ) • v) := continuous_const_smul _ |
| exact hc.measurable.comp |
| ((hfw.comp (hw (ψ*angleNodes m i))).sub (hw (ψ*angleNodes m i))) |
| exact add_error_sq hnm hmid hex.1 hnum.1 hex.2.1 hnum.2 hex.2.2 |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
