| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.FlowRestrict |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory |
| open scoped NNReal RealInnerProductSpace |
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| theorem exists_harmonic_flow {d : ℕ} {H : Point d → ℝ} {L : ℝ≥0} |
| (hH : ContDiff ℝ 2 H) (ht : HasGaussianLowerTail H) |
| (hL : LipschitzWith L (gradient H)) {a b : ℝ} (hab : a≤b) : |
| ∃Ψ : (Point d × Point d) → ℝ → (Point d × Point d), |
| (∀p,Continuous (Ψ p) ∧ Ψ p a=p ∧ ∀t∈Icc a b, |
| HasDerivWithinAt (Ψ p) ((Ψ p t).2,-gradient H (Ψ p t).1) (Icc a b) t) ∧ |
| (∀t∈Icc a b,Continuous (fun p => Ψ p t)) ∧ |
| (∀t∈Icc a b,((gibbs H).prod (stdGaussian (Point d))).map (fun p => Ψ p t)= |
| (gibbs H).prod (stdGaussian (Point d))) := by |
| have hi : Integrable (fun z => Real.exp (-H z)) := by |
| simpa only [inner_zero_left,zero_sub,smul_eq_mul,mul_one] using |
| integrable_tilted H hH.continuous continuous_const ht (growth_const (1:ℝ)) 0 |
| let := probability_gibbs_of_partition (partition_pos_of_continuous hH.continuous).ne' |
| (partition_ne_top_of_integrable hi) |
| let K : Point d → Point d →L[ℝ] Point d := fun _ => -(ContinuousLinearMap.id ℝ (Point d)) |
| let m := fun z => -gradient H z |
| have hK : ContDiff ℝ 1 K := contDiff_const |
| have hm : ContDiff ℝ 1 m := (Trapezoid.contDiff_gradient hH).neg |
| have hKb (y : Point d) : ‖K y‖≤(1:ℝ) := by |
| dsimp [K] |
| rw [norm_neg] |
| exact ContinuousLinearMap.norm_id_le |
| have hmL : LipschitzWith L m := hL.neg |
| have hμ : Integrable (fun z : Point d => ‖z‖) (gibbs H) := by |
| simpa only [pow_one] using (ht.hasExpMoments hH.continuous).hasMoments 1 |
| have hs : ∀u : Point d,∀φ : Point d → ℝ,ContDiff ℝ 1 φ → HasCompactSupport φ → |
| (∫y,inner ℝ u (m y)*φ y ∂gibbs H)=∫y,fderiv ℝ φ y ((K y).adjoint u) ∂gibbs H := by |
| intro u φ hφ hφc |
| have hh := compact_gibbs_ibp (hH.of_le (by norm_num)) hi hφ hφc u |
| dsimp [K,m] |
| simp only [map_neg,ContinuousLinearMap.adjoint_id, |
| neg_apply,ContinuousLinearMap.id_apply, |
| inner_neg_right,neg_mul,integral_neg] |
| simpa only [directional,real_inner_comm] using congrArg Neg.neg hh.symm |
| obtain ⟨Ψ,hΨ,hc,hlaw⟩ := exists_skew_centering_flow hK hm (by norm_num : (0:ℝ)≤1) hKb hmL hμ hs |
| (by norm_num : (0:ℝ)<1) hab |
| refine ⟨Ψ,fun p => ⟨(hΨ p).1,(hΨ p).2.1,fun t ht => ?_⟩,hc,hlaw⟩ |
| simpa only [skewCenteringField,K,m,inv_one,one_smul,neg_apply, |
| ContinuousLinearMap.id_apply,neg_neg,map_neg,ContinuousLinearMap.adjoint_id,neg_smul] using (hΨ p).2.2 t ht |
| 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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