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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 | import Mathlib
import OAI.Probability.LogConcave.Sampling.SkewCenteringField
import OAI.Probability.LogConcave.Sampling.AdjointCoordinateProductLeft
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open scoped Classical BigOperators RealInnerProductSpace
open TensorEnergy
lemma inner_basis_coordinate {d : ℕ} (y : Point d) (i : Fin d) :
inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) y=y i := by
simp [EuclideanSpace.basisFun_apply,PiLp.inner_apply]
theorem jointSkew_field {d : ℕ} {H : Point d → ℝ}
{K : Point d → Point d →L[ℝ] Point d} {m : Point d → Point d}
(hH : Differentiable ℝ H)
(hK : ∀i j,Differentiable ℝ (fun y => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i)
(K y (EuclideanSpace.basisFun (Fin d) ℝ j))))
(hstein : ∀u : Point d,tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ)
(fun i y => inner ℝ u (K y (EuclideanSpace.basisFun (Fin d) ℝ i)))=
fun y => inner ℝ u (m y))
(s : ℝ) (y : Point (d+d)) :
productPointEquiv d d (skewLieField (productPotential H (fun z : Point d => ‖z‖^2/2))
(jointSkew K s) y)=skewCenteringField K m s (productPointEquiv d d y) := by
let L := coordinateProjection (leftCoordinates d d)
have hf (i j : Fin d) : Differentiable ℝ (fun z => inner ℝ
(EuclideanSpace.basisFun (Fin d) ℝ i)
(K (productPointEquiv d d z).1 (EuclideanSpace.basisFun (Fin d) ℝ j))) :=
(hK i j).comp L.differentiable
have hG := (gaussianPotential_polySmooth d).smooth.differentiable (by simp)
apply Prod.ext
· ext z
change skewLieField _ _ y (z.castAdd d)=(-s⁻¹ • (K (productPointEquiv d d y).1).adjoint
(productPointEquiv d d y).2) z
rw [skewLieField_coordinate,Fin.sum_univ_add]
simp only [jointSkew,Sum.elim_inl,Sum.elim_inr,jointSkewEntry_LL,jointSkewEntry_RL,
adjointCoordinate_zero,Finset.sum_const_zero,zero_add]
simp_rw [adjointCoordinate_const_mul (hf _ _)]
simp_rw [adjointCoordinate_product_right hH hG (hK _ _),directional_gaussianPotential]
rw [←Finset.mul_sum]
change (-s⁻¹)*(∑i : Fin d,inner ℝ (productPointEquiv d d y).2
(EuclideanSpace.basisFun (Fin d) ℝ i)*inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i)
(K (productPointEquiv d d y).1 (EuclideanSpace.basisFun (Fin d) ℝ z)))=
(-s⁻¹)*((K (productPointEquiv d d y).1).adjoint (productPointEquiv d d y).2) z
congr 1
rw [←inner_basis_coordinate,ContinuousLinearMap.adjoint_inner_right]
simp_rw [real_inner_comm (productPointEquiv d d y).2,inner_basis_coordinate]
rw [real_inner_comm,PiLp.inner_apply]
simp only [RCLike.inner_apply,conj_trivial]
apply Finset.sum_congr rfl
intro i _
rw [real_inner_comm,inner_basis_coordinate]
· ext z
change skewLieField _ _ y (z.natAdd d)=(s⁻¹ • m (productPointEquiv d d y).1) z
rw [skewLieField_coordinate,Fin.sum_univ_add]
simp only [jointSkew,Sum.elim_inl,Sum.elim_inr,jointSkewEntry_RR,jointSkewEntry_LR,
adjointCoordinate_zero,Finset.sum_const_zero,add_zero]
simp_rw [adjointCoordinate_const_mul (hf _ _)]
simp_rw [adjointCoordinate_product_left hH hG (hK _ _)]
rw [←Finset.mul_sum]
change s⁻¹*tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ)
(fun i w => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ z)
(K w (EuclideanSpace.basisFun (Fin d) ℝ i))) (productPointEquiv d d y).1=_
rw [hstein]
dsimp only
rw [inner_basis_coordinate]
rfl
end LogConcaveSampling
end
end
end
end OAI
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