| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.ReanchorWeight |
| import OAI.Probability.LogConcave.Sampling.ProbabilityTransportTrajectory |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory FinitePicard |
| open scoped Classical BigOperators NNReal |
|
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| lemma FinitePicard.nodes_continuous {I E X : Type*} [Fintype I] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace X] |
| (w : I → I → ℝ) (φ : I → E → E) (a : X → I → E) |
| (hφ : ∀i,Continuous (φ i)) (ha : Continuous a) (N : ℕ) : |
| Continuous (nodes w φ a N) := by |
| induction N with |
| | zero => exact ha |
| | succ N ih => |
| apply continuous_pi |
| intro i |
| exact ((continuous_apply i).comp ha).add (continuous_finsetSum _ (fun j _ => |
| ((hφ j).comp ((continuous_apply j).comp ih)).const_smul (w i j))) |
|
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| lemma probabilityAnchored_continuous {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T h : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (n : ℕ) (hn : 0<n) (N : ℕ) (s i : ProbabilityNode T h n) : |
| Continuous (fun y => probabilityAnchoredPicard F x r T h n N s y i) := by |
| apply (continuous_apply i).comp |
| apply FinitePicard.nodes_continuous |
| · intro j |
| have hj := probabilityNodeTime_mem hT0 hT1 hh hn j |
| exact (probabilityMeanVelocity_lipschitz hF x hr hl hj.1 (hj.2.trans_lt hT1)).continuous |
| · exact continuous_pi (fun _ => continuous_id) |
|
|
| theorem probabilityAnchored_marginal_rms (n : ℕ) (hn : 0<n) : |
| ∃A : ℝ≥0,∃C : ℝ,0≤C ∧ ∃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 → |
| A*probabilityMeanLipschitz lam r≤1/2 → ∀N : ℕ,∀s i : ProbabilityNode T h n, |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let I : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n i,probabilityNodeTime_mem hT0 hT1 hh hn i⟩ |
| Integrable (fun y => ‖probabilityAnchoredPicard F x r T h n N s y i- |
| probabilityTransport hF x hr.le hl hT0.le hT1 S I y‖^2) |
| (interpolationLaw F x r S) ∧ |
| (∫y,‖probabilityAnchoredPicard F x r T h n N s y i- |
| probabilityTransport hF x hr.le hl hT0.le hT1 S I y‖^2 |
| ∂interpolationLaw F x r S)≤ |
| (4*C*((lam:ℝ)*r^2)*Real.sqrt d*(1+Real.log ((d:ℝ)+1))^k*(2*h)^(n+1)+ |
| (A*probabilityMeanLipschitz lam r:ℝ≥0)^N*(2*probabilityInitialRms F x lam r))^2 := by |
| obtain ⟨A,C,hC,k,h⟩ := probabilityAnchored_rms n hn |
| refine ⟨A,C,hC,k,?_⟩ |
| intro d F lam hF x r hr hlam hl hd T hstep hT0 hT1 hh hsmall hq N s i |
| dsimp only |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T hstep n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let I : Icc (0:ℝ) T := ⟨probabilityNodeTime T hstep n i,probabilityNodeTime_mem hT0 hT1 hh hn i⟩ |
| let E := fun y => ‖probabilityAnchoredPicard F x r T hstep n N s y i- |
| probabilityTransport hF x hr.le hl hT0.le hT1 S I y‖^2 |
| have hE : Continuous E := ((probabilityAnchored_continuous hF x hr.le hl hT0 hT1 hh n hn N s i).sub |
| (probabilityTransport_continuous hF x hr.le hl hT0.le hT1 S I)).norm.pow 2 |
| have hm := fullProbabilityFlow_interpolation_closed hF x hr.le hl S.2.1 (S.2.2.trans_lt hT1) |
| have hf := (fullProbabilityFlow_lipschitz hF x hr.le hl ⟨S.2.1,S.2.2.trans hT1.le⟩).continuous.measurable |
| have heq (z : Point d) : E (fullProbabilityFlow hF x hr.le hl S z)= |
| ‖probabilityAnchoredPicard F x r T hstep n N s |
| (fullProbabilityFlow hF x hr.le hl S z) i-fullProbabilityFlow hF x hr.le hl I z‖^2 := by |
| dsimp only [E] |
| rw [probabilityTransport_trajectory] |
| have he := h hF x hr hlam hl hd hT0 hT1 hh hsmall hq N s i |
| change Integrable E (interpolationLaw F x r S) ∧ _ |
| constructor |
| · rw [←hm] |
| apply (integrable_map_measure hE.aestronglyMeasurable hf.aemeasurable).mpr |
| simpa only [Function.comp_def,heq] using he.1 |
| · change (∫y,E y ∂interpolationLaw F x r S)≤_ |
| rw [←hm,integral_map hf.aemeasurable hE.aestronglyMeasurable] |
| simpa only [heq] using he.2 |
|
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| end LogConcaveSampling |
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| end |
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| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory |
| open scoped Classical BigOperators NNReal |
|
|
| theorem harmonicPicard_stationary_rms (n : ℕ) (hn : 0<n) : |
| ∃A : ℝ≥0,∃C : ℝ,1≤C ∧ ∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, |
| Primitive F lam → ∀(x : Point d) {r R T : ℝ}, |
| 0<r → 0<lam → (lam:ℝ)*r^2≤1/2 → 0<R → R^2≤1-T^2 → 0≤T → T<1 → |
| ∀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 T) (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,(gibbs (centeringPotential F x r T)).map (fun y => Ξ y v)= |
| gibbs (centeringPotential F x r T)) → |
| ∀z : ℝ,0≤z → z≤1 → A*probabilityMeanLipschitz lam r≤1/2 → |
| ∀N : ℕ,∀i : Fin (n+1), |
| Integrable (fun y => ‖harmonicPicard F x r T z n N (productPointEquiv d d y) i- |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n i))).1‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖harmonicPicard F x r T z n N (productPointEquiv d d y) i- |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n i))).1‖^2 |
| ∂gibbs (centeringPotential F x r T))≤ |
| (2*(r*T)*Real.sqrt (4*C^2*(z^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(R⁻¹)^(4*(n+1))*harmonicMeanBudget (n+1)))+ |
| (A*probabilityMeanLipschitz lam r:ℝ≥0)^N*(r*T)* |
| Real.sqrt (harmonicStationaryMoment F x r T))^2 := by |
| obtain ⟨A,hA⟩ := harmonicWeight_budget n hn |
| obtain ⟨C,hC,hErr⟩ := harmonic_mean_quadrature_rms n hn |
| refine ⟨A,C,hC,?_⟩ |
| intro d F lam hF x r R T hr hlam hl hR hRT hT0 hT1 Ξ hc hinit hder hm hlaw z hz hz1 hq N i |
| have hH := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) |
| have ht := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) (gaussianPotential_lowerTail d) |
| let : IsProbabilityMeasure (gibbs (centeringPotential F x r T)) := |
| probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| let B := 4*C^2*(z^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(R⁻¹)^(4*(n+1))*harmonicMeanBudget (n+1)) |
| have hB : 0≤B := by dsimp [B]; positivity [harmonicMeanBudget_nonneg (n+1)] |
| let D := (r*T)*Real.sqrt B |
| let H := (r*T)*Real.sqrt (harmonicStationaryMoment F x r T) |
| have hD : 0≤D := by dsimp [D]; positivity |
| have hH0 : 0≤H := by dsimp [H]; positivity |
| have ha (j : Fin (n+1)) : MemLp (fun y : Point (d+d) => |
| harmonicFree n z (productPointEquiv d d y) j) 2 (gibbs (centeringPotential F x r T)) := by |
| apply centering_lipschitz_memLp hF x hr hlam hl hT0 hT1 |
| exact (LipschitzWith.eval j).comp ((harmonicFree_lipschitz n z).comp (productPointEquiv d d).lipschitzWith) |
| have hi (j : Fin (n+1)) : z*probabilityNodes n j∈Icc (0:ℝ) 1 := |
| ⟨mul_nonneg hz (probabilityNodes_mem hn j).1, |
| (mul_le_of_le_one_right hz (probabilityNodes_mem hn j).2).trans hz1⟩ |
| have he (j : Fin (n+1)) : MemLp (fun y : Point (d+d) => |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1) 2 |
| (gibbs (centeringPotential F x r T)) := by |
| have hproj := centering_lipschitz_memLp hF x hr hlam hl hT0 hT1 |
| (LipschitzWith.prod_fst.comp (productPointEquiv d d).lipschitzWith) |
| exact hproj.comp_measurePreserving ⟨hm.comp (measurable_const.prodMk measurable_id),hlaw _ (hi j)⟩ |
| have hdef (j : Fin (n+1)) : |
| (∫y,‖FinitePicard.step (harmonicWeight n z) (fun _ => harmonicMeanVelocity F x r T) |
| (harmonicFree n z (productPointEquiv d d y)) |
| (fun j => (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1) j- |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1‖^2 |
| ∂gibbs (centeringPotential F x r T))≤D^2 := by |
| simp_rw [harmonicPicard_exact_defect hF x hr.le hl hT0 hT1 Ξ hc hinit hder hz hz1 n hn, |
| norm_smul,mul_pow,Real.norm_eq_abs,sq_abs] |
| rw [integral_const_mul] |
| dsimp only [D] |
| simp only [mul_pow,Real.sq_sqrt hB] |
| exact mul_le_mul_of_nonneg_left |
| (hErr hF x hr hlam hl hR hRT hT0 hT1 Ξ hc hder hm hlaw z hz hz1 j).2 |
| (mul_nonneg (sq_nonneg r) (sq_nonneg T)) |
| have hstart (j : Fin (n+1)) : |
| (∫y,‖harmonicFree n z (productPointEquiv d d y) j- |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1‖^2 |
| ∂gibbs (centeringPotential F x r T))≤H^2 := by |
| dsimp only [H] |
| rw [mul_pow,Real.sq_sqrt (harmonicStationaryMoment_nonneg F x r T)] |
| exact (harmonicFree_initial_rms hF x hr hlam hl hT0 hT1 Ξ hc hinit hder hm hlaw hz hz1 n hn j).2 |
| have hout := FinitePicard.nodes_rms (A:=A) (harmonicWeight n z) |
| (fun _ => harmonicMeanVelocity F x r T) |
| (fun _ => harmonicMeanVelocity_lipschitz hF x hr.le hl hT0 hT1) |
| (fun j y => harmonicFree n z (productPointEquiv d d y) j) |
| (fun j y => (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1) |
| ha he (hA z hz hz1) hq hD hH0 hdef hstart N i |
| simpa only [harmonicPicard,FinitePicard.nodes_eq_iterate,D,H,B,mul_assoc] using hout |
| 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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