| import Mathlib |
| import OAI.Analysis.BiholderTransport.Contact.HopfSupports |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle Metric |
| open scoped Topology ContDiff NNReal |
|
|
| namespace WeakMTWTransport |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty 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] |
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|
| omit [CompactSpace M] [Nonempty M] in |
| lemma cost_divided_triangle_two_times (x y z : M) {s t : ℝ} (hs : 0<s) (ht : 0<t) : |
| cost x z/(s+t) ≤ cost x y/s+cost y z/t := by |
| have hst : 0<s+t := add_pos hs ht |
| have ha : 0<s/(s+t) := div_pos hs hst |
| have ha1 : s/(s+t)<1 := (div_lt_one hst).mpr (lt_add_of_pos_right s ht) |
| have H := div_le_div_of_nonneg_right (divided_cost_triangle x y z ha ha1) hst.le |
| calc |
| _ ≤ _ := H |
| _ = _ := by |
| have he : 1-s/(s+t)=t/(s+t) := by field_simp; ring |
| rw [he] |
| field_simp |
|
|
| omit [CompactSpace M] [Nonempty 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] in |
| lemma mdifferentiableAt_of_chart_differentiable {f : M → ℝ} {x : M} |
| (hd : DifferentiableAt ℝ (fun y => f ((extChartAt 𝓘(ℝ,Model n) x).symm y)) |
| (extChartAt 𝓘(ℝ,Model n) x x)) : |
| MDifferentiableAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) f x := by |
| have H := hd.hasFDerivAt.hasMFDerivAt.comp x |
| (mdifferentiableAt_extChartAt (I := 𝓘(ℝ,Model n)) |
| (show x∈(chartAt (Model n) x).source from mem_chart_source _ _)).hasMFDerivAt |
| apply (H.congr_of_eventuallyEq ?_).mdifferentiableAt |
| filter_upwards [extChartAt_source_mem_nhds (I := 𝓘(ℝ,Model n)) x] with y hy |
| simp only [Function.comp_apply,(extChartAt 𝓘(ℝ,Model n) x).left_inv hy] |
|
|
| lemma WeakMTW.mdifferentiable_hopfLax (hmtw : WeakMTW (n := n) (M := M)) |
| {u v : M → ℝ} (hu : Continuous u) (hv : Continuous v) (hdual : IsCostDualPair u v) |
| {t : ℝ} (ht : 0<t) (ht1 : t<1) : |
| MDifferentiable 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (hopfLax t u) := by |
| intro z |
| obtain ⟨B,hB,r,hr,_,H⟩ := hmtw.uniform_hopfLax_C11 z |
| apply mdifferentiableAt_of_chart_differentiable |
| exact (H t ht ht1 u v hu hv hdual).1 _ (mem_ball_self hr) |
|
|
| lemma hopfLax_derivative_of_minimizer {u : M → ℝ} (hu : Continuous u) |
| {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} (hp : p∈minimizingVectors x) |
| {t : ℝ} (ht : 0<t) (ht1 : t≤1) |
| (heq : hopfLax t u (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 = |
| u x+cost x (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1/t) |
| (hd : MDifferentiableAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (hopfLax t u) |
| (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) : |
| HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (hopfLax t u) |
| (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 |
| (innerSL ℝ (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2) := by |
| let w : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,p⟩ |
| let z := sprayFlow t w |
| obtain ⟨ε,hε,hεt,_,_,Hε⟩ := exists_two_short_legs x p ht |
| (show t<t+1 by linarith) zero_lt_one |
| have hshort := (Filter.Eventually.self_of_nhds Hε).2.2.2 |
| let a := (sprayFlow (t-ε) w).1 |
| let G : M → ℝ := fun y => u x+cost x a/(t-ε)+cost a y/ε |
| have hG : HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) G z.1 (innerSL ℝ z.2) := by |
| have H := (hasMFDerivAt_const (I := 𝓘(ℝ,Model n)) (I' := 𝓘(ℝ,ℝ)) |
| (u x+cost x a/(t-ε)) z.1).add (hshort.const_smul ε⁻¹) |
| have heD : 0+ε⁻¹ • innerSL ℝ (ε • (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2) = |
| innerSL ℝ z.2 := by |
| ext v |
| simp only [zero_add,_root_.smul_apply,smul_eq_mul,innerSL_apply_apply, |
| real_inner_smul_left] |
| dsimp [z,w] |
| field_simp |
| replace H := H.congr_mfderiv heD |
| have heG : G = ((fun _ => u x+cost x a/(t-ε)) + |
| ε⁻¹ • fun y => dist (sprayFlow (t-ε) (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 y^2/2) := by |
| funext y |
| dsimp [G,cost,a,w] |
| ring |
| rw [heG] |
| exact H |
| have hmin : dist x z.1=t*‖p‖ := by |
| dsimp [z,w] |
| rw [←riemannianExp_smul,minimizingVectors_smul hp ht.le ht1,norm_smul, |
| Real.norm_eq_abs,abs_of_pos ht] |
| have hleft : dist x a=(t-ε)*‖p‖ := |
| sprayFlow_minimizing_prefix w (sub_pos.mpr hεt).le (by linarith) hmin |
| have hright : dist a z.1=ε*‖p‖ := by |
| have H := sprayFlow_minimizing_subinterval w (sub_pos.mpr hεt).le (by linarith) hmin |
| simpa only [sub_sub_cancel] using H |
| have hcontact : hopfLax t u z.1=G z.1 := by |
| rw [heq] |
| dsimp [G,cost] |
| rw [hmin,hleft,hright] |
| field_simp |
| ring |
| have hle : ∀ y, hopfLax t u y≤G y := by |
| intro y |
| have H := cost_divided_triangle_two_times x a y (sub_pos.mpr hεt) hε |
| rw [sub_add_cancel] at H |
| exact (hopfLax_le hu t y x).trans (by dsimp [G]; linarith) |
| exact hd.hasMFDerivAt.congr_mfderiv |
| ((upper_contact_mfderiv hd hG.mdifferentiableAt (Filter.Eventually.of_forall hle) hcontact).trans hG.mfderiv) |
| |
| def riemannianGradient (f : M → ℝ) (x : M) : TangentSpace 𝓘(ℝ,Model n) x := by |
| letI : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) |
| exact (InnerProductSpace.toDual ℝ (TangentSpace 𝓘(ℝ,Model n) x)).symm |
| (mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) f x) |
| |
| omit [CompactSpace M] [Nonempty M] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] in |
| lemma riemannianGradient_eq_of_derivative {f : M → ℝ} {x : M} |
| {v : TangentSpace 𝓘(ℝ,Model n) x} |
| (h : HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) f x (innerSL ℝ v)) : |
| riemannianGradient (n := n) f x=v := by |
| let : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) |
| rw [riemannianGradient,h.mfderiv] |
| exact (InnerProductSpace.toDual ℝ (TangentSpace 𝓘(ℝ,Model n) x)).symm_apply_apply v |
| |
| lemma WeakMTW.hopfLax_backward_flow (hmtw : WeakMTW (n := n) (M := M)) |
| {v : M → ℝ} (hv : Continuous v) {t : ℝ} (ht : 0<t) (ht1 : t<1) |
| (q : subgradientGraph (n := n) (cTransform v)) : |
| sprayFlow (-t) |
| (⟨graphProjection (cTransform v) t q, |
| riemannianGradient (hopfLax t (cTransform v)) (graphProjection (cTransform v) t q)⟩ : |
| TangentBundle 𝓘(ℝ,Model n) M) = q.1 := by |
| have hdual : IsCostDualPair (cTransform v) (cTransform (cTransform v)) := |
| ⟨(cTransform_triple hv).symm,rfl⟩ |
| have hu := continuous_cTransform hv |
| have hp := (hmtw.global_support hv q.2).1 |
| have hbase : graphProjection (cTransform v) t q=(sprayFlow t q.1).1 := |
| riemannianExp_smul _ _ _ |
| have hd := hmtw.mdifferentiable_hopfLax hu (continuous_cTransform hu) hdual ht ht1 |
| have hval := hmtw.hopfLax_graph_value hv ht ht1 q |
| rw [hbase] at hval |
| have hder := hopfLax_derivative_of_minimizer hu hp ht ht1.le hval (hd _) |
| rw [hbase,riemannianGradient_eq_of_derivative hder] |
| change sprayFlow (-t) (sprayFlow t q.1)=q.1 |
| rw [←sprayFlow_add,neg_add_cancel,sprayFlow_zero] |
| |
| end WeakMTWTransport |
| |
| end |
| |
| end OAI |
| |
