| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.TransportCorrectionMajorant |
|
|
| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory TensorEnergy Quadrature |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
|
|
| local instance centeringStateBudgetDecidableEqUnit : DecidableEq Unit := Classical.decEq _ |
|
|
| lemma centeringSkew_inv_scaled {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T s : 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) (hs : 0<s) : |
| ∀k y,AllSplitBound |
| (spatialTensor (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (List.finRange k) y) |
| ((2*kernelMajorant k)*((lam:ℝ)*r/s)*(R⁻¹)^k) := by |
| intro k y |
| convert centering_jointSkew_spatial_bound hF x hr hlam hl hR hRT hT0 hT1 s y using 1 |
| rw [abs_of_pos (inv_pos.mpr hs)] |
| simp only [inv_pow,div_eq_mul_inv] |
| ring |
|
|
| lemma identityArray_polySmooth (d : ℕ) (c : Unit → Fin d) : |
| PolySmooth (vectorArray (id : Point d → Point d) c) := |
| vectorArray_polySmooth contDiff_id LipschitzWith.id (fun _ => 1) |
| (fun _ => by norm_num) (fun _ _ y => multilinearTensor_id_bound y) c |
|
|
| lemma identityArray_inv_energy {d k : ℕ} {H : Point d → ℝ} |
| (hH : Continuous H) (ht : HasGaussianLowerTail H) {R : ℝ} |
| (hR : 0<R) (hR1 : R≤1) (hk : 0<k) : |
| spatialEnergy (vectorArray (id : Point d → Point d)) k (gibbs H)≤ |
| d*(R⁻¹)^(2*k)*1 := by |
| have he := vectorArray_energy hH ht contDiff_id (identityArray_polySmooth d) hk |
| (fun y => multilinearTensor_id_bound (n:=k) y) |
| simp only [one_pow,mul_one] at he ⊢ |
| apply he.trans |
| exact le_mul_of_one_le_right (Nat.cast_nonneg _) (one_le_pow₀ ((one_le_inv₀ hR).mpr hR1)) |
|
|
| def centeringStateBudget (n : ℕ) : ℝ := |
| iterEnergyBudget harmonicGradientBound (fun k => 2*kernelMajorant k) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) (fun _ => 1) n 0 |
|
|
| def centeringMeanBudget (n : ℕ) : ℝ := |
| iterEnergyBudget harmonicGradientBound (fun k => 2*kernelMajorant k) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) |
| (fun k => (normalizedTensorMajorant (k+1) 1)^2) n 0 |
|
|
| theorem centering_state_taylor_uniform {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T s : 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) (hs : 0<s) |
| (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) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (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)) : |
| let A := vectorArray (id : Point (d+d) → Point (d+d)) |
| ∀n : ℕ,∀a b : ℝ,0≤a → a≤b → b≤1 → |
| Integrable (fun y => ‖tensorVector A (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) A k) (Ξ y t)) n a b‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖tensorVector A (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) A k) (Ξ y t)) n a b‖^2 |
| ∂gibbs (centeringPotential F x r T))≤ |
| ((b-a)^(n+1)/(n.factorial:ℝ))^2* |
| ((d+d)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)^(n+1)*centeringStateBudget (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 A 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 q := stationary_lie_taylor_rms (S:=Unit) (d:=d+d) hH ht |
| (centeringPotential_gradient_uniform hF x hr hl hT0 hT1) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) A |
| (centeringSkew_polySmooth hF x hr hlam hl hT0 hT1 s) (identityArray_polySmooth (d+d)) |
| (jointSkew_skew _ s) (fun k => 2*kernelMajorant k) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) (fun _ => 1) |
| (α:=(lam:ℝ)*r/s) (κ:=d+d) (ρ:=R⁻¹) (by positivity) ((one_le_inv₀ hR).mpr hR1) |
| (fun _ => by norm_num) (centeringSkew_inv_scaled hF x hr hlam hl hR hRT hT0 hT1 hs) |
| (centeringScore_inv_scaled hF x hr hlam hl hR hRT hT0 hT1) |
| (fun k hk => by simpa only [Nat.cast_add] using identityArray_inv_energy hH.smooth.continuous ht hR hR1 hk) |
| Ξ hder hm hlaw hab hsub n |
| simpa only [centeringStateBudget,Nat.cast_add] using q |
|
|
| theorem centering_mean_taylor_uniform {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T s : 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) (hs : 0<s) |
| (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) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (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) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) |
| (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) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) |
| (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)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)^(n+1)*centeringMeanBudget (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⟩ |
| exact stationary_lie_taylor_rms (S:=Unit) (d:=d+d) hH ht |
| (centeringPotential_gradient_uniform hF x hr hl hT0 hT1) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (centeringMeanArray F x r T) |
| (centeringSkew_polySmooth hF x hr hlam hl hT0 hT1 s) |
| (centeringMeanArray_polySmooth hF x hr hlam hl hT0 hT1) |
| (jointSkew_skew _ s) (fun k => 2*kernelMajorant k) |
| (fun k => 2+normalizedTensorMajorant (k+1) 1) |
| (fun k => (normalizedTensorMajorant (k+1) 1)^2) |
| (α:=(lam:ℝ)*r/s) (κ:=d*((lam:ℝ)*r)^2) (ρ:=R⁻¹) (by positivity) ((one_le_inv₀ hR).mpr hR1) |
| (fun k => sq_nonneg _) (centeringSkew_inv_scaled hF x hr hlam hl hR hRT hT0 hT1 hs) |
| (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 |
| end LogConcaveSampling |
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| end |
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| end |
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| end OAI |
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