| import Mathlib |
| import OAI.Analysis.BiholderTransport.Regularity.MetricDefinitions |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ActualRadial |
| import OAI.Analysis.BiholderTransport.Coordinates.SplitAction |
| import OAI.Analysis.BiholderTransport.Regularity.SecondScaled |
| import OAI.Analysis.BiholderTransport.CostGeometry.BranchCost2 |
| import OAI.Analysis.BiholderTransport.Coordinates.ProperPieces |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| section Crossing |
| 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)) |
| end Crossing |
| end WeakMTWTransport |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| section MiddleHessian |
| 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)) |
|
|
| omit [IsRiemannianManifold 𝓘(ℝ,Model n) M] in |
| lemma reverseRay_prefix (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) : |
| reverseRay (tangentScale h z)=tangentScale (-h) (sprayFlow h z) := by |
| rw [reverseRay,sprayFlow_scale,mul_one,tangentScale_mul,neg_one_mul] |
|
|
| lemma proper_prefix_reverse_in_injectivityDomain {z : TangentBundle 𝓘(ℝ,Model n) M} |
| (hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) : |
| (-h) • (sprayFlow h z).2∈injectivityDomain (sprayFlow h z).1 := by |
| have H := reverseRay_injectivityDomain (z := tangentScale h z) |
| (contracted_minimizer_mem_injectivityDomain hz hh hh1) |
| rw [reverseRay_prefix] at H |
| exact H |
|
|
| lemma proper_prefix_reverse_endpoint (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) : |
| riemannianExp (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)=z.1 := by |
| have H := reverseRay_endpoint (tangentScale h z) |
| rw [reverseRay_prefix] at H |
| exact H |
|
|
| def middleHessian (z : TangentBundle 𝓘(ℝ,Model n) M) (h T : ℝ) : |
| TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1 →L[ℝ] |
| TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1 →L[ℝ] ℝ := |
| h⁻¹ • normalHessian (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)+ |
| (T-h)⁻¹ • normalHessian (sprayFlow h z).1 ((T-h) • (sprayFlow h z).2) |
|
|
| omit [IsRiemannianManifold 𝓘(ℝ,Model n) M] in |
| lemma middleHessian_apply (z : TangentBundle 𝓘(ℝ,Model n) M) (h T : ℝ) |
| (v w : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : |
| middleHessian z h T v w= |
| normalHessian (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) v w/h+ |
| normalHessian (sprayFlow h z).1 ((T-h) • (sprayFlow h z).2) v w/(T-h) := by |
| simp only [middleHessian,add_apply,smul_apply, |
| smul_eq_mul,div_eq_mul_inv] |
| ring |
|
|
| def middleCenteredAction (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) |
| (b : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : ℝ := |
| h⁻¹*(normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) b- |
| normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) 0)- |
| (-(1-h)⁻¹)*(normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) b- |
| normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) 0) |
|
|
| lemma middle_normal_prefix (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) |
| (b : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : |
| normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) b= |
| cost z.1 (riemannianExp (sprayFlow h z).1 b) := by |
| unfold normalCost |
| rw [proper_prefix_reverse_endpoint,cost_symm] |
|
|
| lemma middle_normal_suffix (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) |
| (b : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : |
| normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) b= |
| cost (riemannianExp (sprayFlow h z).1 b) (riemannianExp z.1 z.2) := by |
| unfold normalCost |
| rw [proper_suffix_endpoint,one_smul] |
|
|
| lemma middle_action_contact {z : TangentBundle 𝓘(ℝ,Model n) M} |
| (hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) : |
| normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) 0/h+ |
| normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) 0/(1-h)= |
| cost z.1 (riemannianExp z.1 z.2) := by |
| rw [middle_normal_prefix,middle_normal_suffix,riemannianExp_zero] |
| have H := splitNormalAction_contact hz hh hh1 |
| simp only [splitNormalAction,normalCost,riemannianExp_zero] at H |
| simpa only [riemannianExp_smul] using H |
|
|
| lemma middleCenteredAction_minimum {z : TangentBundle 𝓘(ℝ,Model n) M} |
| (hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) : |
| IsLocalMin (middleCenteredAction z h) 0 := by |
| filter_upwards [] with b |
| have H := squared_dist_divided_action_le z.1 (riemannianExp (sprayFlow h z).1 b) |
| (riemannianExp z.1 z.2) hh hh1 |
| have H' : cost z.1 (riemannianExp z.1 z.2)≤ |
| normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) b/h+ |
| normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) b/(1-h) := by |
| rw [middle_normal_prefix,middle_normal_suffix] |
| convert H using 1 <;> simp only [cost,div_div,mul_comm] |
| rw [←middle_action_contact hz hh hh1] at H' |
| dsimp only [middleCenteredAction] |
| simp only [sub_self,mul_zero,neg_mul,sub_neg_eq_add,div_eq_mul_inv] at H' ⊢ |
| nlinarith |
| |
| lemma middleCenteredAction_contDiffAt {z : TangentBundle 𝓘(ℝ,Model n) M} |
| (hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) : |
| ContDiffAt ℝ 2 (middleCenteredAction z h) 0 := by |
| have hf : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)) 0 := |
| (normalCost_contDiffAt (proper_prefix_reverse_in_injectivityDomain hz hh hh1)).of_le |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) |
| have hg : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2)) 0 := |
| (normalCost_contDiffAt (proper_suffix_in_injectivityDomain hz hh hh1 le_rfl)).of_le |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) |
| exact (contDiffAt_const.mul (hf.sub contDiffAt_const)).sub |
| (contDiffAt_const.mul (hg.sub contDiffAt_const)) |
| |
| lemma middleCenteredAction_hessian {z : TangentBundle 𝓘(ℝ,Model n) M} |
| (hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) |
| (v : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : |
| fderiv ℝ (fderiv ℝ (middleCenteredAction z h)) 0 v v=middleHessian z h 1 v v := by |
| have hf : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)) 0 := |
| (normalCost_contDiffAt (proper_prefix_reverse_in_injectivityDomain hz hh hh1)).of_le |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) |
| have hg : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2)) 0 := |
| (normalCost_contDiffAt (proper_suffix_in_injectivityDomain hz hh hh1 le_rfl)).of_le |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) |
| unfold middleCenteredAction |
| rw [second_fderiv_scaled_differences hf hg,middleHessian_apply] |
| simp only [normalHessian,neg_mul,sub_neg_eq_add,div_eq_mul_inv,mul_comm] |
| |
| lemma middleHessian_one_nonneg {z : TangentBundle 𝓘(ℝ,Model n) M} |
| (hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) |
| (v : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : |
| 0 ≤ middleHessian z h 1 v v := by |
| rw [←middleCenteredAction_hessian hz hh hh1] |
| exact localMin_hessian_nonneg (middleCenteredAction_contDiffAt hz hh hh1) |
| (middleCenteredAction_minimum hz hh hh1) v |
| |
| end MiddleHessian |
| end WeakMTWTransport |
| |
| end |
| |
| end |
| |
| end |
| |
| end OAI |
| |
