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import OAI.Probability.LogConcave.JetEstimates.NormalizedLaplace
import OAI.Probability.LogConcave.Sampling.PrimitiveField
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
namespace LogConcaveSampling
open MeasureTheory
open scoped RealInnerProductSpace
def jointPosition (d : ℕ) : Point d →L[ℝ] JointPoint d :=
EuclideanSpace.sumEquivProd.symm.toContinuousLinearMap.comp (ContinuousLinearMap.inl ℝ _ _)
def jointField (d : ℕ) : Point d →L[ℝ] JointPoint d :=
EuclideanSpace.sumEquivProd.symm.toContinuousLinearMap.comp (ContinuousLinearMap.inr ℝ _ _)
lemma jointPair_inner {d : ℕ} (u v w z : Point d) :
inner ℝ (EuclideanSpace.sumEquivProd.symm (u,v) : JointPoint d)
(EuclideanSpace.sumEquivProd.symm (w,z)) = inner ℝ u w+inner ℝ v z := by
have he := jointImage_norm_sq (u+w) (v+z)
rw [show (u+w,v+z)=(u,v)+(w,z) from rfl,map_add,norm_add_sq_real] at he
rw [jointImage_norm_sq,jointImage_norm_sq,norm_add_sq_real,norm_add_sq_real] at he
linarith
lemma jointPosition_inner {d : ℕ} (u : Point d) (f : Point d → Point d) (L : ℝ) (z : Point d) :
inner ℝ (jointPosition d u) (jointImage f L z)=inner ℝ u z := by
change inner ℝ (EuclideanSpace.sumEquivProd.symm (u,0) : JointPoint d)
(EuclideanSpace.sumEquivProd.symm (z,L⁻¹ • f z))=_
rw [jointPair_inner,inner_zero_left,add_zero]
lemma jointField_inner {d : ℕ} (u : Point d) (f : Point d → Point d) (L : ℝ) (z : Point d) :
inner ℝ (jointField d u) (jointImage f L z)=L⁻¹*inner ℝ u (f z) := by
change inner ℝ (EuclideanSpace.sumEquivProd.symm (0,u) : JointPoint d)
(EuclideanSpace.sumEquivProd.symm (z,L⁻¹ • f z))=_
rw [jointPair_inner,inner_zero_left,zero_add,inner_smul_right]
lemma jointPosition_basis {d : ℕ} (i : Fin d) :
jointPosition d ((EuclideanSpace.basisFun (Fin d) ℝ) i)=
(EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ) (Sum.inl i) := by
simp only [EuclideanSpace.basisFun_apply]
ext j
cases j with
| inl j =>
change (EuclideanSpace.single i (1:ℝ)) j=(EuclideanSpace.single (Sum.inl i : Fin d ⊕ Fin d) (1:ℝ)) (Sum.inl j)
simp [PiLp.single_apply]
| inr j =>
change (0:ℝ)=(EuclideanSpace.single (Sum.inl i : Fin d ⊕ Fin d) (1:ℝ)) (Sum.inr j)
simp
lemma jointField_basis {d : ℕ} (i : Fin d) :
jointField d ((EuclideanSpace.basisFun (Fin d) ℝ) i)=
(EuclideanSpace.basisFun (Fin d ⊕ Fin d) ℝ) (Sum.inr i) := by
simp only [EuclideanSpace.basisFun_apply]
ext j
cases j with
| inl j =>
change (0:ℝ)=(EuclideanSpace.single (Sum.inr i : Fin d ⊕ Fin d) (1:ℝ)) (Sum.inl j)
simp
| inr j =>
change (EuclideanSpace.single i (1:ℝ)) j=(EuclideanSpace.single (Sum.inr i : Fin d ⊕ Fin d) (1:ℝ)) (Sum.inr j)
simp [PiLp.single_apply]
lemma joint_laplace_one {d : ℕ} {μ : Measure (Point d)} {f : Point d → Point d}
(hf : Continuous f) (L : ℝ) (θ : Point d) :
Appell.laplace (μ.map (jointImage f L)) (fun _ => (1:ℝ)) (jointPosition d θ)=
Appell.laplace μ (fun _ => (1:ℝ)) θ := by
have hS : Continuous (jointImage f L) := by
unfold jointImage
fun_prop
rw [Appell.laplace,integral_map hS.aemeasurable (by fun_prop)]
simp only [jointPosition_inner,Appell.laplace]
lemma joint_normalizedLaplace {d : ℕ} {μ : Measure (Point d)} {f : Point d → Point d}
(hf : Continuous f) (L : ℝ) (u θ : Point d) :
Appell.normalizedLaplace (μ.map (jointImage f L))
(fun z => inner ℝ (jointField d u) z) (jointPosition d θ)=
L⁻¹*Appell.normalizedLaplace μ (fun z => inner ℝ u (f z)) θ := by
have hS : Continuous (jointImage f L) := by
unfold jointImage
fun_prop
have hn : Appell.laplace (μ.map (jointImage f L))
(fun z => inner ℝ (jointField d u) z) (jointPosition d θ)=
L⁻¹*Appell.laplace μ (fun z => inner ℝ u (f z)) θ := by
rw [Appell.laplace,integral_map hS.aemeasurable (by fun_prop)]
simp only [jointPosition_inner,jointField_inner,smul_eq_mul]
rw [show (fun z : Point d => Real.exp (inner ℝ θ z)*(L⁻¹*inner ℝ u (f z)))=
fun z => L⁻¹*(Real.exp (inner ℝ θ z)*inner ℝ u (f z)) by funext z; ring,
integral_const_mul,Appell.laplace]
rfl
unfold Appell.normalizedLaplace
rw [joint_laplace_one hf,hn]
ring
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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