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import OAI.Probability.LogConcave.Sampling.TransportCorrectionMajorant
namespace OAI
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
end
end
end OAI
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