File size: 10,899 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 | import Mathlib
import OAI.Probability.LogConcave.Sampling.ConditionalMeanArray
import OAI.Probability.LogConcave.Dynamics.ActualJointCenteringFlow
import OAI.Probability.LogConcave.Sampling.StationaryTensorVectorRms
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 gradient_gaussianPotential {d : ℕ} (y : Point d) :
gradient (fun z : Point d => ‖z‖^2/2) y=y := by
ext i
rw [←directional_basis_gradient,directional_gaussianPotential]
simp [EuclideanSpace.basisFun_apply,PiLp.inner_apply]
lemma gaussianPotential_gradient_lipschitz (d : ℕ) :
LipschitzWith 1 (gradient (fun z : Point d => ‖z‖^2/2)) := by
have he : gradient (fun z : Point d => ‖z‖^2/2)=id := funext gradient_gaussianPotential
rw [he]
exact LipschitzWith.id
def centeringPotential {d : ℕ} (F : Point d → ℝ) (x : Point d) (r T : ℝ) : Point (d+d) → ℝ :=
productPotential (interpolationPotential F x r T) (fun z : Point d => ‖z‖^2/2)
def centeringMeanArray {d : ℕ} (F : Point d → ℝ) (x : Point d) (r T : ℝ) :
(Unit → Fin (d+d)) → Point (d+d) → ℝ :=
arrayCoordinateLift (leftCoordinates d d) (fun _ => leftCoordinates d d)
(conditionalMeanArray F x r T)
def centeringGradientBound (lam : ℝ≥0) (r T : ℝ) (h : 0≤(Real.pi^2/2)*T*((lam:ℝ)*r)) : ℝ≥0 :=
(1+‖r*T‖₊*⟨(Real.pi^2/2)*T*((lam:ℝ)*r),h⟩)+1
lemma centeringPotential_gradient_lipschitz {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r)
(hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) :
LipschitzWith (centeringGradientBound lam r T (by positivity))
(gradient (centeringPotential F x r T)) :=
productPotential_gradient_lipschitz
((interpolationPotential_smooth hF x hr.le hl hT0 hT1).differentiable (by simp))
((gaussianPotential_polySmooth d).smooth.differentiable (by simp))
(interpolationPotential_gradient_lipschitz hF x hr.le hl hT0 hT1)
(gaussianPotential_gradient_lipschitz d)
lemma centeringMeanArray_polySmooth {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)
(c : Unit → Fin (d+d)) : PolySmooth (centeringMeanArray F x r T c) :=
arrayCoordinateLift_polySmooth _ _ _ (conditionalMeanArray_polySmooth hF x hr hlam hl hT0 hT1) c
lemma centeringMeanArray_energy {d n : 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)
(hn : 0<n) :
spatialEnergy (centeringMeanArray F x r T) n (gibbs (centeringPotential F x r T))≤
d*((lam:ℝ)*r)^2*(normalizedTensorMajorant (n+1) (1-T^2))^2 := by
have hH := interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1
have ht := interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1
let := probability_gibbs_of_gaussianTail hH.smooth.continuous ht
dsimp only [centeringMeanArray,centeringPotential]
rw [spatialEnergy_leftLift hH.smooth.continuous _
(fun c => (conditionalMeanArray_polySmooth hF x hr hlam hl hT0 hT1 c).smooth)]
exact conditionalMeanArray_energy hF x hr hlam hl hT0 hT1 hn
lemma centeringSkew_polySmooth {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) (s : ℝ) :
∀c,PolySmooth (jointSkew (centeringKernel hF x hr hl hT0 hT1) s c) := by
apply jointSkew_polySmooth
intro i j
simpa only [matrixArray_apply,Sum.elim_inl,Sum.elim_inr] using
centeringKernel_polySmooth hF x hr hlam hl hT0 hT1 (Sum.elim (fun _ => i) (fun _ => j))
lemma centeringSkew_scaled_bound {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/R^k)*((lam:ℝ)*r/s)*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 [one_pow,mul_one]
ring
lemma centeringScore_scaled_bound {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) :
∀k,0<k → ∀y,AllSplitBound
(spatialTensor (scoreField (centeringPotential F x r T)) (List.finRange k) y)
((2+normalizedTensorMajorant (k+1) (1-T^2))*1^k) := by
intro k hk y
simpa only [centeringPotential,one_pow,mul_one] using
centeringScore_allSplit hF x hr hlam hl hT0 hT1 hk y
lemma centeringMean_scaled_energy {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) :
∀k,0<k → spatialEnergy (centeringMeanArray F x r T) k
(gibbs (centeringPotential F x r T))≤
(d*((lam:ℝ)*r)^2)*1^(2*k)*(normalizedTensorMajorant (k+1) (1-T^2))^2 := by
intro k hk
simpa only [one_pow,mul_one] using centeringMeanArray_energy hF x hr hlam hl hT0 hT1 hk
theorem centering_mean_taylor_rms {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)^(n+1)*
iterEnergyBudget (centeringGradientBound lam r T (by positivity))
(fun k => 2*kernelMajorant k/R^k)
(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)
(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/R^k)
(fun k => 2+normalizedTensorMajorant (k+1) (1-T^2))
(fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2)
(α:=(lam:ℝ)*r/s) (κ:=d*((lam:ℝ)*r)^2) (ρ:=1) (by positivity) le_rfl
(fun _ => sq_nonneg _)
(centeringSkew_scaled_bound hF x hr hlam hl hR hRT hT0 hT1 hs)
(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_centering_mean_taylor_rms {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),
(∀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)
(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)^(n+1)*
iterEnergyBudget (centeringGradientBound lam r T (by positivity))
(fun k => 2*kernelMajorant k/R^k)
(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_actual_joint_centering_flow hF x hr hlam hl hT0 hT1 hs
exact ⟨Ξ,hinit,hm,hlaw,centering_mean_taylor_rms hF x hr hlam hl hR hRT hT0 hT1 hs Ξ hder hm hlaw⟩
end LogConcaveSampling
end
end
end
end OAI
|