| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Model | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section ModelMixedConorm | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteModelMixed (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end ModelMixedConorm | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section UniformMixedConorm | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteUniformConorm (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end UniformMixedConorm | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff NNReal | |
| namespace WeakMTWTransport | |
| section UniformRadialProfile | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteUniformProfile (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end UniformRadialProfile | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section AffineMTW | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteAffine (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end AffineMTW | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff NNReal | |
| namespace WeakMTWTransport | |
| section UniformJoinSpectrum | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteJoinBound (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end UniformJoinSpectrum | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section UniformAffineProfile | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteAffineProfile (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end UniformAffineProfile | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section AffineSpectrumComparison | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteAffineSpectrum (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end AffineSpectrumComparison | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section FixedJoinDeterminant | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| local instance tangentFiniteJoinDet (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end FixedJoinDeterminant | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Module | |
| namespace WeakMTWTransport | |
| section NormDetBound | |
| variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| [FiniteDimensional ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] | |
| [FiniteDimensional ℝ F] | |
| lemma norm_eigenvector_image_eq_singularValue (A : E →ₗ[ℝ] F) | |
| {n : ℕ} (hn : finrank ℝ E=n) (i : Fin n) : | |
| ‖A (A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn i)‖=A.singularValues i := by | |
| let v := A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn i | |
| have hv : ‖v‖=1 := (A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn).norm_eq_one i | |
| have he : (A.adjoint.comp A) v=A.isSymmetric_adjoint_comp_self.eigenvalues hn i • v := | |
| A.isSymmetric_adjoint_comp_self.apply_eigenvectorBasis hn i | |
| have hsq : ‖A v‖^2=A.isSymmetric_adjoint_comp_self.eigenvalues hn i := by | |
| rw [←real_inner_self_eq_norm_sq,←A.adjoint_inner_left v (A v)] | |
| change inner ℝ ((A.adjoint.comp A) v) v=_ | |
| rw [he,real_inner_smul_left,real_inner_self_eq_norm_sq,hv] | |
| ring | |
| apply (sq_eq_sq₀ (norm_nonneg _) (A.singularValues_nonneg i)).mp | |
| rw [hsq,A.sq_singularValues_fin hn i] | |
| end NormDetBound | |
| end WeakMTWTransport | |
| end | |
| end | |
| end | |
| end OAI | |