| import Mathlib |
| 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 |
|
|
