| 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 |
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| end OAI |
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