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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import Mathlib
import OAI.Analysis.BiholderTransport.Geodesics.UniformFixedCrossing
import OAI.Analysis.BiholderTransport.Calculus.SamplePoleDerivative
import OAI.Analysis.BiholderTransport.Regularity.NonnegativeTransfer
namespace OAI
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
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]
omit [Nonempty M] in
lemma exists_actual_uniform_regular_split_after {t:ℝ} (ht : 0<t) (ht1 : t<1) :
∃ s:ℝ,t<s ∧ s<1 ∧ ∀ x:M,∀ p:TangentSpace 𝓘(ℝ,Model n) x,
p∈minimizingVectors x → s • p∈injectivityDomain x ∧
(1-s) • (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).2∈
injectivityDomain (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 := by
have H : ∀ᶠ s in 𝓝[<] (1:ℝ),t<s ∧ s<1 ∧ ∀ x:M,
∀ p:TangentSpace 𝓘(ℝ,Model n) x,‖p‖≤Metric.diam (univ:Set M) →
(1-s) • (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).2∈
injectivityDomain (sprayFlow s (⟨x,p⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 := by
filter_upwards [(uniformly_shifted_injectivityDomain_near_one
(n := n) (M := M) (Metric.diam (univ:Set M))).filter_mono nhdsWithin_le_nhds,
(eventually_gt_nhds ht1).filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin]
with s hs hst hs1
exact ⟨hst,hs1,hs⟩
obtain ⟨s,hst,hs1,hs⟩ := H.exists
exact ⟨s,hst,hs1,fun x p hp=>⟨contracted_minimizer_mem_injectivityDomain hp
(ht.trans hst) hs1,hs x p (minimizing_norm_le_diam hp)⟩⟩
lemma WeakMTW.actual_sample_pole_matrix (hmtw : WeakMTW (n := n) (M := M))
{v : M → ℝ} (hv : Continuous v) {t : ℝ} (ht : 0<t) (ht1 : t<1)
{x a c : M} {p : TangentSpace 𝓘(ℝ,Model n) x}
(hp : p∈normalSubdifferential (n := n) (cTransform v) x)
(hz : riemannianExp x (t • p)∈(extChartAt 𝓘(ℝ,Model n) a).source)
(hx : x∈(extChartAt 𝓘(ℝ,Model n) c).source)
(hD : DifferentiableAt ℝ
(fun w : Model n => extChartAt 𝓘(ℝ,Model n) c
(hopfPole (n := n) t (cTransform v) ((extChartAt 𝓘(ℝ,Model n) a).symm w)))
(extChartAt 𝓘(ℝ,Model n) a (riemannianExp x (t • p)))) :
let V := TangentSpace 𝓘(ℝ,Model n) x
let I : V →L[ℝ] V →L[ℝ] ℝ := innerSL ℝ
let L : V →L[ℝ] V →L[ℝ] ℝ := t • I - fderiv ℝ (fderiv ℝ
(fun q:V=>hopfLax t (cTransform v) (riemannianExp x (t • q)))) p
DifferentiableAt ℝ (fderiv ℝ
(fun q:V=>hopfLax t (cTransform v) (riemannianExp x (t • q)))) p ∧
(∀ d:V,0≤L d d) ∧ (∀ d e:V,L d e=L e d) ∧
(∀ q∈activeLogs v x,∀ d:V,L (q-p) d=0) ∧
(∀ (ι:Type) [Fintype ι] (pj:ι → V) (w:ι → ℝ),
(∀ j,0≤w j) → (∑ j,w j=1) → (∑ j,w j • pj j=p) →
(∀ j,pj j∈activeLogs v x) → ∀ i,0<w i →
HasLowerSecondTaylor (fun d:V=>v (riemannianExp x (pj i+d))+‖pj i+d‖^2/2)
0 (w i • L)) := by
dsimp only
let V := TangentSpace 𝓘(ℝ,Model n) x
let : NormedAddCommGroup (V →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup
let : NormedSpace ℝ (V →L[ℝ] ℝ) := ContinuousLinearMap.toNormedSpace
let : NormedAddCommGroup (V →L[ℝ] V →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup
let q : subgradientGraph (n := n) (cTransform v) := ⟨⟨x,p⟩,hp⟩
have hpole : hopfPole (n := n) t (cTransform v) (riemannianExp x (t • p))=x := by
exact congrArg Bundle.TotalSpace.proj (hmtw.hopfLax_backward_flow hv ht ht1 q)
obtain ⟨e,R,he,he0,herep⟩ := hmtw.exists_actual_normal_pole_derivative hv ht ht1 hz hx hpole hD
have hpre := hmtw.active_hull_precut hv ht ht1 x p
(normalSubdifferential_subset_active_hull hv x hp)
have hdual : IsCostDualPair (cTransform v) (cTransform (cTransform v)) :=
⟨(cTransform_triple hv).symm,rfl⟩
have HD' := hmtw.actualPole_second_derivative (continuous_cTransform hv)
(continuous_cTransform (continuous_cTransform hv)) hdual ht ht1 hpre he he0 herep
have HD := HD'.fderiv
rw [HD,sub_sub_cancel]
have H := hmtw.actualPole_taylor (continuous_cTransform hv)
(continuous_cTransform (continuous_cTransform hv)) hdual ht ht1 hpre he he0 herep
obtain ⟨s,hts,hs1,hs⟩ := exists_actual_uniform_regular_split_after (n := n) (M := M) ht ht1
refine ⟨HD'.differentiableAt,H.2.1,?_,?_,?_⟩
· intro d f
change inner ℝ (R d) f=inner ℝ (R f) d
exact (H.2.2 d f).trans (real_inner_comm (R f) d)
· intro r hr d
have hk := hmtw.actualPole_kernel (continuous_cTransform hv)
(continuous_cTransform (continuous_cTransform hv)) hdual ht ht1 (ht.trans hts) hs1
hpre hr.1 (active_contact_biconjugate hv hr.2).2 (hs x r hr.1).1 (hs x r hr.1).2
he he0 herep
simp only [ContinuousLinearMap.comp_apply,hk,map_zero,zero_apply]
· intro ι _ pj w hw hw1 hbar ha i hi
exact hmtw.modified_active_lower_jet_nonneg hv ht hts hs1 pj w hw hw1 i hi
(fun j=>(ha j).1) (fun j=>(ha j).2) (fun j=>(hs x (pj j) (ha j).1).1)
(fun j=>(hs x (pj j) (ha j).1).2) (hbar ▸ he) (hbar ▸ he0) (hbar ▸ herep)
end WeakMTWTransport
end
end OAI
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