| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.TransportCorrectionMajorant |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
| open TensorEnergy Quadrature |
|
|
| local instance : DecidableEq Unit := Classical.decEq _ |
|
|
| lemma jointCorrectionArray_polySmooth {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T : ℝ} |
| (hr : 0<r) (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) |
| (hR : 0<R) (hRT : R^2≤1-T^2) (hT0 : 0≤T) (hT1 : T<1) |
| (s t : Icc (0:ℝ) T) (c : Unit → Fin (d+d)) : |
| PolySmooth (jointCorrectionArray |
| (fun z => probabilityTransport hF x hr.le hl hT0 hT1 s t z-z) c) := |
| arrayCoordinateLift_polySmooth _ _ _ |
| (probabilityCorrection_polySmooth hF x hr hlam hl hR hRT hT0 hT1 s t) c |
|
|
| def harmonicCorrectionBudget (n : ℕ) : ℝ := |
| iterEnergyBudget harmonicGradientBound (fun _ => 2) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) |
| (fun k => (transportCorrectionMajorant k)^2) n 0 |
|
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| lemma harmonicCorrectionBudget_nonneg (n : ℕ) : 0≤harmonicCorrectionBudget n := |
| iterEnergyBudget_nonneg harmonicGradientBound.2 _ _ (fun _ => sq_nonneg _) n 0 |
|
|
| def harmonicMeanBudget (n : ℕ) : ℝ := |
| iterEnergyBudget harmonicGradientBound (fun _ => 2) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) |
| (fun k => (normalizedTensorMajorant (k+1) 1)^2) n 0 |
|
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| lemma harmonicMeanBudget_nonneg (n : ℕ) : 0≤harmonicMeanBudget n := |
| iterEnergyBudget_nonneg harmonicGradientBound.2 _ _ (fun _ => sq_nonneg _) n 0 |
|
|
| theorem harmonic_correction_taylor_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T : ℝ} |
| (hr : 0<r) (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) |
| (hR : 0<R) (hRT : R^2≤1-T^2) (hT0 : 0≤T) (hT1 : T<1) |
| (s t u : Icc (0:ℝ) T) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) |
| (hder : ∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r u) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v) |
| (hm : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1)) |
| (hlaw : ∀v∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r u)).map (fun y => Ξ y v)= |
| gibbs (centeringPotential F x r u)) : |
| let A := jointCorrectionArray (fun z => probabilityTransport hF x hr.le hl hT0 hT1 s t z-z) |
| ∀n : ℕ,∀a b : ℝ,0≤a → a≤b → b≤1 → |
| Integrable (fun y => ‖tensorVector A (Ξ y b)-chainTaylor |
| (fun k v => tensorVector (iterTensorLie (centeringPotential F x r u) |
| (harmonicSkew d) A k) (Ξ y v)) n a b‖^2) (gibbs (centeringPotential F x r u)) ∧ |
| (∫y,‖tensorVector A (Ξ y b)-chainTaylor |
| (fun k v => tensorVector (iterTensorLie (centeringPotential F x r u) |
| (harmonicSkew d) A k) (Ξ y v)) n a b‖^2 ∂gibbs (centeringPotential F x r u))≤ |
| ((b-a)^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r^2)^2)*(R⁻¹)^(4*(n+1))*harmonicCorrectionBudget (n+1)) := by |
| have hu1 := u.2.2.trans_lt hT1 |
| have hRu : R^2≤1-(u:ℝ)^2 := by nlinarith [u.2.1,u.2.2] |
| have hR1 : R≤1 := by nlinarith [sq_nonneg T] |
| have hH : PolySmooth (centeringPotential F x r u) := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl u.2.1 hu1) (gaussianPotential_polySmooth d) |
| have ht : HasGaussianLowerTail (centeringPotential F x r u) := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) u.2.1 hu1) (gaussianPotential_lowerTail d) |
| let := probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| intro A n a b ha hab hb |
| have hsub : uIcc a b⊆Icc (0:ℝ) 1 := by |
| rw [uIcc_of_le hab] |
| intro v h; exact ⟨ha.trans h.1,h.2.trans hb⟩ |
| have h := stationary_lie_taylor_rms (S:=Unit) (d:=d+d) hH ht |
| (centeringPotential_gradient_uniform hF x hr hl u.2.1 hu1) |
| (harmonicSkew d) A (harmonicSkew_polySmooth d) |
| (jointCorrectionArray_polySmooth hF x hr hlam hl hR hRT hT0 hT1 s t) |
| (jointSkew_skew _ 1) (fun _ => 2) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) |
| (fun k => (transportCorrectionMajorant k)^2) |
| (α:=1) (κ:=d*((lam:ℝ)*r^2)^2) (ρ:=R⁻¹) (by positivity) ((one_le_inv₀ hR).mpr hR1) |
| (fun _ => sq_nonneg _) (harmonicSkew_inv_scaled hR hR1) |
| (centeringScore_inv_scaled hF x hr hlam hl hR hRu u.2.1 hu1) |
| (jointCorrection_inv_scaled hF x hr hlam hl hR hRT hT0 hT1 s t u) |
| Ξ hder hm hlaw hab hsub n |
| simpa only [one_pow,one_mul,←pow_mul,harmonicCorrectionBudget] using h |
|
|
| theorem harmonic_mean_taylor_uniform {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T : ℝ} |
| (hr : 0<r) (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) |
| (hR : 0<R) (hRT : R^2≤1-T^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)*(R⁻¹)^(4*(n+1))*harmonicMeanBudget (n+1)) := by |
| have hR1 : R≤1 := by nlinarith [sq_nonneg T] |
| 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) hH ht |
| (centeringPotential_gradient_uniform 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) |
| (fun k => (normalizedTensorMajorant (k+1) 1)^2) |
| (α:=1) (κ:=d*((lam:ℝ)*r)^2) (ρ:=R⁻¹) (by positivity) ((one_le_inv₀ hR).mpr hR1) |
| (fun _ => sq_nonneg _) (harmonicSkew_inv_scaled hR hR1) |
| (centeringScore_inv_scaled hF x hr hlam hl hR hRT hT0 hT1) |
| (centeringMean_inv_scaled hF x hr hlam hl hR hRT hT0 hT1) |
| Ξ hder hm hlaw hab hsub n |
| simpa only [one_pow,one_mul,←pow_mul,harmonicMeanBudget] using h |
| end LogConcaveSampling |
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| end |
|
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| end |
|
|
| end |
|
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| end OAI |
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|
