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import OAI.Probability.LogConcave.Numerics.ReanchorWeight
import OAI.Probability.LogConcave.Sampling.ProbabilityTransportTrajectory
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory ProbabilityTheory FinitePicard
open scoped Classical BigOperators NNReal
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)))
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
end LogConcaveSampling
end
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
end
end
end
end OAI
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