| import Mathlib |
| import OAI.Probability.LogConcave.Analysis.CenteringPotential |
| import OAI.Probability.LogConcave.Dynamics.JointHarmonicFlow |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
| open TensorEnergy Quadrature |
|
|
| local instance harmonicMeanTaylorRmsDecidableEqUnit : DecidableEq Unit := Classical.decEq _ |
|
|
| theorem harmonic_mean_taylor_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) |
| (hder : ∀y t,t∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r T) |
| (harmonicSkew d) (Ξ y t)) (Icc (0:ℝ) 1) t) |
| (hm : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1)) |
| (hlaw : ∀t∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r T)).map (fun y => Ξ y t)= |
| gibbs (centeringPotential F x r T)) : |
| ∀n : ℕ,∀a b : ℝ,0≤a → a≤b → b≤1 → |
| Integrable (fun y => ‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (harmonicSkew d) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (harmonicSkew d) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2 |
| ∂gibbs (centeringPotential F x r T))≤ |
| ((b-a)^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)* |
| iterEnergyBudget (centeringGradientBound lam r T (by positivity)) |
| (fun _ => 2) |
| (fun k => 2+normalizedTensorMajorant (k+1) (1-T^2)) |
| (fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2) (n+1) 0) := by |
| have hH : PolySmooth (centeringPotential F x r T) := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) |
| have ht : HasGaussianLowerTail (centeringPotential F x r T) := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) (gaussianPotential_lowerTail d) |
| let := probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| intro n a b ha hab hb |
| have hsub : uIcc a b⊆Icc (0:ℝ) 1 := by |
| rw [uIcc_of_le hab] |
| intro t h; exact ⟨ha.trans h.1,h.2.trans hb⟩ |
| have h := stationary_lie_taylor_rms (S:=Unit) (d:=d+d) |
| (H:=centeringPotential F x r T) hH ht |
| (centeringPotential_gradient_lipschitz hF x hr hl hT0 hT1) |
| (harmonicSkew d) (centeringMeanArray F x r T) |
| (harmonicSkew_polySmooth d) (centeringMeanArray_polySmooth hF x hr hlam hl hT0 hT1) |
| (jointSkew_skew _ 1) |
| (fun _ => 2) |
| (fun k => 2+normalizedTensorMajorant (k+1) (1-T^2)) |
| (fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2) |
| (α:=1) (κ:=d*((lam:ℝ)*r)^2) (ρ:=1) (by positivity) le_rfl |
| (fun _ => sq_nonneg _) |
| (harmonicSkew_scaled_bound d) |
| (centeringScore_scaled_bound hF x hr hlam hl hT0 hT1) |
| (centeringMean_scaled_energy hF x hr hlam hl hT0 hT1) |
| Ξ hder hm hlaw hab hsub n |
| refine ⟨h.1,h.2.trans ?_⟩ |
| simp only [one_pow,mul_one] |
| exact le_rfl |
|
|
| theorem exists_harmonic_mean_taylor_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) : |
| ∃Ξ : Point (d+d) → ℝ → Point (d+d), |
| (∀y,Continuous (Ξ y) ∧ Ξ y 0=y) ∧ |
| Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1) ∧ |
| (∀t∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r T)).map (fun y => Ξ y t)= |
| gibbs (centeringPotential F x r T)) ∧ |
| ∀n : ℕ,∀a b : ℝ,0≤a → a≤b → b≤1 → |
| Integrable (fun y => ‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (harmonicSkew d) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (harmonicSkew d) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2 |
| ∂gibbs (centeringPotential F x r T))≤ |
| ((b-a)^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)* |
| iterEnergyBudget (centeringGradientBound lam r T (by positivity)) |
| (fun _ => 2) |
| (fun k => 2+normalizedTensorMajorant (k+1) (1-T^2)) |
| (fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2) (n+1) 0) := by |
| obtain ⟨Ξ,hinit,hder,hm,hlaw⟩ := exists_joint_harmonic_flow |
| ((interpolationPotential_smooth hF x hr.le hl hT0 hT1).of_le (by norm_cast)) |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) |
| (interpolationPotential_gradient_lipschitz hF x hr.le hl hT0 hT1) |
| exact ⟨Ξ,hinit,hm,hlaw,harmonic_mean_taylor_rms hF x hr hlam hl hT0 hT1 Ξ hder hm hlaw⟩ |
| 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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