openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /LinearAlgebra /EndpointSpectrumBound.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Regularity.FiniteIntervalMesh | |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ShearedSpectrum | |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ConvexSpectrum | |
| import OAI.Analysis.BiholderTransport.Regularity.SingularValuesBound | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section EndpointSpectrumBound | |
| 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 (x : M) : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| lemma uniform_endpoint_singular_bound : | |
| ∃ D : ℝ, 0<D ∧ ∀ x : M, ∀ p : TangentSpace 𝓘(ℝ,Model n) x, | |
| p∈minimizingVectors x → ∀ i : Fin (Module.finrank ℝ (Model n)), | |
| (endpointExpDifferential x p).toLinearMap.singularValues i ≤ D := by | |
| obtain ⟨a,b,ha,hb,H⟩ := uniform_radial_singular_shortening (n := n) (M := M) | |
| obtain ⟨l,u,hl,hu,J⟩ := uniform_split_exp_bounds (n := n) (M := M) | |
| refine ⟨u/a,div_pos hu ha,?_⟩ | |
| intro x p hp i | |
| have hj := (J (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M) hp (1/2) | |
| (by constructor <;> norm_num)).1 | |
| have hj' : ∀ v : TangentSpace 𝓘(ℝ,Model n) x, | |
| ‖endpointExpDifferential x ((1/2:ℝ) • p) v‖ ≤ u*‖v‖ := fun v => by | |
| exact (hj v).2 | |
| have hS := singularValues_le_of_bound (E := TangentSpace 𝓘(ℝ,Model n) x) | |
| (F := TangentSpace 𝓘(ℝ,Model n) (riemannianExp x ((1/2:ℝ) • p))) (endpointExpDifferential x ((1/2:ℝ) • p)).toLinearMap | |
| hu.le hj' rfl i | |
| have hR := (H x p hp (1/2) (by constructor <;> norm_num) i).1 | |
| rw [show (1:ℝ)-1/2=1/2 by norm_num] at hR | |
| have HH := hR.trans hS | |
| rw [le_div_iff₀ ha] | |
| nlinarith only [HH,ha] | |
| end EndpointSpectrumBound | |
| end WeakMTWTransport | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle ContinuousLinearMap | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section AffineSpectrum | |
| 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 (x : M) : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| lemma WeakMTW.uniform_extended_spectral_comparison | |
| (hmtw : WeakMTW (n := n) (M := M)) {a₀ B : ℝ} (ha₀ : 0<a₀) (hB : 0<B) : | |
| ∃ c : ℝ, 0<c ∧ ∀ x : M, ∀ p q : TangentSpace 𝓘(ℝ,Model n) x, | |
| 0 < inner ℝ p q → ‖q‖^2/inner ℝ p q ≤ B → | |
| (∀ t∈Icc (-a₀) 1, p+t • q∈minimizingVectors x) → | |
| ∀ i : Fin (Module.finrank ℝ (Model n)), | |
| c*(endpointExpDifferential x (p+q)).toLinearMap.singularValues i ≤ | |
| (endpointExpDifferential x p).toLinearMap.singularValues i := by | |
| let a := min a₀ (1/(2*B)) | |
| have ha : 0<a := lt_min ha₀ (div_pos (by norm_num) (by positivity)) | |
| have haB : a*B ≤ 1/2 := by | |
| have HH := min_le_right a₀ (1/(2*B)) | |
| have HH2 := (le_div_iff₀ (show 0<2*B by positivity)).mp HH | |
| nlinarith only [HH2] | |
| obtain ⟨l,u,b,hl,hu,hb,HSp⟩ := uniform_sheared_radial_spectrum (n := n) (M := M) ha hB haB | |
| obtain ⟨C,hC,HB⟩ := uniform_sheared_radial_quadratic_bounds (n := n) (M := M) ha hB haB | |
| obtain ⟨D,hD,HD⟩ := uniform_endpoint_singular_bound (n := n) (M := M) | |
| let e := min (1/2:ℝ) (l/(64*C)) | |
| have he0 : 0<e := lt_min (by norm_num) (div_pos hl (by positivity)) | |
| have he1 : e<1 := (min_le_left _ _).trans_lt (by norm_num) | |
| have heC : e*C ≤ l/32 := by | |
| have HH := (le_div_iff₀ (show 0<64*C by positivity)).mp | |
| (min_le_right (1/2:ℝ) (l/(64*C))) | |
| change e*(64*C) ≤ l at HH | |
| nlinarith only [HH,hl] | |
| let d := min a e | |
| have hdpos : 0<d := lt_min ha he0 | |
| let K := 2*C/d | |
| have hK : 0 ≤ K := div_nonneg (by positivity) hdpos.le | |
| obtain ⟨c,hc,HC⟩ := uniform_finite_profile_comparison hC hl hu hb hD ⟨he0,he1⟩ heC hK | |
| refine ⟨c,hc,?_⟩ | |
| intro x p q hpq hβ hline₀ j | |
| have hline (t : ℝ) (ht : t∈Icc (-a) 1) : p+t • q∈minimizingVectors x := | |
| hline₀ t ⟨by have := min_le_left a₀ (1/(2*B)); change a ≤ a₀ at this; linarith [ht.1],ht.2⟩ | |
| let i := j.rev | |
| let s : ℝ → ℝ := fun t => (endpointExpDifferential x (p+t • q)).toLinearMap.singularValues i.rev | |
| have hSym (δ : ℝ) (hd : δ∈Ioc (0:ℝ) (1/2)) (t : ℝ) (ht : t∈Icc (-a) 1) : | |
| (shearedRadialOperator x p q δ t).toLinearMap.IsSymmetric := | |
| shearedRadialOperator_symmetric (contracted_minimizer_mem_injectivityDomain (hline t ht) | |
| (by linarith [hd.2]) (by linarith [hd.1])) | |
| let f : ℝ → ℝ → ℝ := fun δ t => | |
| if H : δ∈Ioc (0:ℝ) (1/2) ∧ t∈Icc (-a) 1 then (hSym δ H.1 t H.2).eigenvalues rfl i else 0 | |
| have hf (δ : ℝ) (hd : δ∈Ioc (0:ℝ) (1/2)) (t : ℝ) (ht : t∈Icc (-a) 1) : | |
| f δ t=(hSym δ hd t ht).eigenvalues rfl i := by | |
| dsimp only [f] | |
| rw [dite_eq_left (show δ∈Ioc (0:ℝ) (1/2) ∧ t∈Icc (-a) 1 from ⟨hd,ht⟩)] | |
| have hI (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : t∈Icc (-a) 1 := ⟨by linarith [ht.1],ht.2⟩ | |
| have hSmall (t : ℝ) (ht : t∈Icc (0:ℝ) (1-e)) : t∈Icc (0:ℝ) 1 := | |
| ⟨ht.1,by linarith [ht.2]⟩ | |
| have hInt (t : ℝ) (ht : t∈Icc (0:ℝ) (1-e)) : t∈Icc (-a+d) (1-d) := by | |
| have hda : d ≤ a := min_le_left _ _ | |
| have hde : d ≤ e := min_le_right _ _ | |
| constructor <;> linarith [ht.1,ht.2] | |
| have h0 : (0:ℝ)∈Icc (-a) 1 := ⟨by linarith,by norm_num⟩ | |
| have h1 : (1:ℝ)∈Icc (-a) 1 := ⟨by linarith,le_rfl⟩ | |
| have hm : 1-e∈Icc (-a) 1 := ⟨by linarith,by linarith⟩ | |
| have HS (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : 0 ≤ s t := | |
| (endpointExpDifferential x (p+t • q)).toLinearMap.singularValues_nonneg i.rev | |
| have HD' : s 1 ≤ D := HD x (p+1 • q) (hline 1 h1) i.rev | |
| have HP (δ : ℝ) (hd : δ∈Ioc (0:ℝ) (1/2)) (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : | |
| l*δ/(s t+δ)-b*δ ≤ f δ t ∧ f δ t ≤ u*δ/(s t+δ)+b*δ := by | |
| obtain ⟨hF,H⟩ := HSp x p q hpq hβ hline δ hd t (hI t ht) | |
| rw [hf δ hd t (hI t ht)] | |
| exact H i | |
| have HL (δ : ℝ) (hd : δ∈Ioc (0:ℝ) (1/2)) (r : ℝ) (hr : r∈Icc (0:ℝ) (1-e)) | |
| (t : ℝ) (ht : t∈Icc (0:ℝ) (1-e)) : |f δ t-f δ r| ≤ K*|t-r| := by | |
| rw [hf δ hd t (hI t (hSmall t ht)),hf δ hd r (hI r (hSmall r hr))] | |
| exact bounded_matrix_convex_eigen_secant (hSym δ hd) | |
| (hmtw.shearedRadialOperator_convexOn x p q hpq.ne' hd hline) hC.le hdpos | |
| (fun t ht v => (HB x p q hpq hβ hline δ hd t ht v).2) | |
| (hInt r hr) (hInt t ht) (hI r (hSmall r hr)) (hI t (hSmall t ht)) rfl i | |
| have HE (δ : ℝ) (hd : δ∈Ioc (0:ℝ) (1/2)) : f δ (1-e) ≤ (1-e)*f δ 1+e*C := by | |
| rw [hf δ hd (1-e) hm,hf δ hd 1 h1] | |
| exact matrix_convex_eigen_endpoint ha.le ⟨he0.le,he1.le⟩ (hSym δ hd) | |
| (hmtw.shearedRadialOperator_convexOn x p q hpq.ne' hd hline) | |
| (fun v => (le_abs_self _).trans (HB x p q hpq hβ hline δ hd 0 h0 v).2) hm h1 rfl i | |
| have result := HC s f HS HD' HP HL HE | |
| have hz : p+(0:ℝ) • q=p := by simp | |
| have ho : p+(1:ℝ) • q=p+q := by simp | |
| have hsz : s 0=(endpointExpDifferential x p).toLinearMap.singularValues j := by | |
| change (endpointExpDifferential x (p+(0:ℝ) • q)).toLinearMap.singularValues i.rev=_ | |
| rw [hz,show i.rev=j from Fin.rev_rev j] | |
| have hso : s 1=(endpointExpDifferential x (p+q)).toLinearMap.singularValues j := by | |
| change (endpointExpDifferential x (p+(1:ℝ) • q)).toLinearMap.singularValues i.rev=_ | |
| rw [ho,show i.rev=j from Fin.rev_rev j] | |
| rwa [hsz,hso] at result | |
| end AffineSpectrum | |
| end WeakMTWTransport | |
| end | |
| end | |
| end | |
| end OAI | |