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e5a97c6 | 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 | import OAI.Combinatorics.MatchingPSD.TableGeometry
namespace OAI
namespace MatchingSmoothing
open scoped BigOperators
attribute [local instance] Classical.propDecidable
lemma simultaneous_cross_approximation (K : ℕ) :
∃ d H : ℕ, 0 < d ∧ ∀ h : ℕ, H ≤ h →
∀ (V G I T : Type) [Fintype V] [Fintype G] [Group G] [Fintype I] [Fintype T]
(_e : V ≃ Vertices h) (z : I → V → Fin 3), (∀ i c, 0 < histogram (z i) c) →
∀ (M : I → T → Pairing V)
(lift : I → Equiv.Perm V → G)
(g : I → G → Pairing V → ℝ),
(∀ i a p, g i (a * lift i p) = fun N => g i a (Pairing.relabel p N)) →
∀ B : ℝ, (∀ i a N, (g i a N)^2 ≤ B^2) →
∃ a : G, ∃ P : I → (Fin 3 → ℝ) → ℝ,
(∀ i, P i ∈ lowDegree (fun c (v : Fin 3 → ℝ) => v c) d) ∧
(∑ i, ∑ t : T,
(predicateAverage (fun N => crossVector (z i) N = crossVector (z i) (M i t)) (g i a) -
P i (crossVector (z i) (M i t)))^2)
≤ (Fintype.card I : ℝ) * Fintype.card T * ((2*h : ℝ)^(2*K))⁻¹ * B^2 := by
classical
obtain ⟨d,H,hd,hmain⟩ := simultaneous_table_approximation K
refine ⟨d,H,hd,?_⟩
intro h hh V G I T _ _ _ _ _ e z hz M lift g hcov B hB
let ev := Pairing.transportEquiv e.symm
let z' := fun i => z i ∘ e.symm
let M' := fun i t => Pairing.transport e (M i t)
let lift' := fun i (p : Equiv.Perm (Vertices h)) => lift i (e.trans (p.trans e.symm))
let g' := fun i a N => g i a (ev N)
have hc (i : I) (a : G) (p : Equiv.Perm (Vertices h)) :
g' i (a * lift' i p) = fun N => g' i a (Pairing.relabel p N) := by
funext N
change g i (a * lift' i p) (ev N) = _
rw [hcov]
change g i a (Pairing.relabel (e.trans (p.trans e.symm)) (Pairing.transport e.symm N)) =
g i a (Pairing.transport e.symm (Pairing.relabel p N))
rw [Pairing.transport_relabel]
rfl
obtain ⟨a,P,hP,hbound⟩ := hmain h hh G I T z'
(fun i c => by
have he : histogram (z' i) = histogram (z i) := histogram_comp_equiv (z i) e.symm
rw [he]
exact hz i c)
M' lift' g' hc B (fun i a N => hB i a (ev N))
refine ⟨a,P,hP,?_⟩
have hvec (i : I) (t : T) : crossVector (z' i) (M' i t) = crossVector (z i) (M i t) := by
rw [crossVector_transport]
have he : z' i ∘ e = z i := by funext v; simp [z']
rw [he]
have hav (i : I) (t : T) :
predicateAverage (fun N => N.table (z' i) = (M' i t).table (z' i)) (g' i a) =
predicateAverage (fun N => crossVector (z i) N = crossVector (z i) (M i t)) (g i a) := by
apply predicateAverage_equiv ev
intro N
rw [table_eq_iff_cross, hvec]
change _ ↔ crossVector (z i) (Pairing.transport e.symm N) = _
rw [crossVector_transport]
have htvec (i : I) (t : T) :
(fun c => ((M' i t).table (z' i) (threeEdgeType (crossPair c)) : ℝ)) =
crossVector (z i) (M i t) := by
rw [← hvec]
funext c
exact (crossCoordinate_eq_table _ _ _).symm
simpa only [hav, htvec] using hbound
end MatchingSmoothing
end OAI
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