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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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import OAI.Combinatorics.MatchingPSD.MatrixInterpolation
namespace OAI
namespace MatchingInterpolation
open MatchingSmoothing PerfectMatchingPSD Matrix
open scoped BigOperators
attribute [local instance] Classical.propDecidable
abbrev SampleIndex (d : ℕ) := Fin 5 ⊕ GridIndex d
noncomputable def samplePoint (d : ℕ) (L : ℝ) : SampleIndex d → Point :=
Sum.elim specialPoint (scaledNode L)
lemma specialPoint_explicit (j : Fin 5) :
specialPoint j = (![![1,0,0], ![0,1,1], ![3,0,0], ![1,2,0], ![1,0,2]] j : Point) := by
fin_cases j <;> ext a <;> fin_cases a <;> norm_num [specialPoint, origin, dirP, dirStar, dirA, dirB]
lemma sample_witness {s l d : ℕ} (hs : 1 ≤ s) (hsl : 2*l*d ≤ s) (t : SampleIndex d) :
∃ M : Pairing (TripleVertices s), crossVector threeColor M = samplePoint d (2*l) t := by
rcases t with j | g
· rw [samplePoint, Sum.elim_inl, specialPoint_explicit]
exact triple_pairing_five hs j
· exact triple_pairing_grid hsl g
lemma good_cell_polynomials (K : ℕ) :
∃ d H : ℕ, 0 < d ∧ ∀ s q l r : ℕ, H ≤ 3*s+1 → 1 ≤ s → 2*l*d ≤ s →
∀ (B : Pairing (GlobalVertices s q) → Matrix (Fin r) (Fin r) ℝ),
(∀ M, (B M).PosSemidef) → (∀ M, (B M).trace ≤ (r:ℝ)) →
∀ η : ℝ, 0 ≤ η →
(q:ℝ)*r^4*(Fintype.card (SampleIndex d))*((2*(3*s+1):ℝ)^(2*K))⁻¹ ≤ η^2 →
∃ a : Equiv.Perm (GlobalVertices s q),
∃ P : Fin q → Fin r → Fin r → Point → ℝ,
(∀ i u v, P i u v ∈ lowDegree coord d) ∧
∀ i u v t, |cellMatrix i (fun M => B (Pairing.relabel a M))
(samplePoint d (2*l) t) u v-P i u v (samplePoint d (2*l) t)| ≤ η := by
classical
obtain ⟨d,H,hd,hmain⟩ := simultaneous_cross_approximation K
refine ⟨d,H,hd,?_⟩
intro s q l r hH hs hsl B hB htr η hη hnum
let J := Fin q × (Fin r × Fin r)
let T := SampleIndex d
let M : T → Pairing (TripleVertices s) := fun t => (sample_witness hs hsl t).choose
have hM (t : T) : crossVector threeColor (M t) = samplePoint d (2*l) t :=
(sample_witness hs hsl t).choose_spec
let g (j : J) (a : Equiv.Perm (GlobalVertices s q)) (N : Pairing (TripleVertices s)) : ℝ :=
extensionAverage j.1 (fun Z => B (Pairing.relabel a Z) j.2.1 j.2.2) N
have hg (j : J) (a : Equiv.Perm (GlobalVertices s q)) (p : Equiv.Perm (TripleVertices s)) :
g j (a * liftLocal j.1 p) = fun N => g j a (Pairing.relabel p N) := by
funext N
dsimp only [g]
rw [extensionAverage_relabel]
simp only [Pairing.relabel_mul]
have hgb (j : J) (a : Equiv.Perm (GlobalVertices s q)) (N : Pairing (TripleVertices s)) :
(g j a N)^2 ≤ (r:ℝ)^2 :=
extensionAverage_sq_bound j.1 _ (fun Z => hB _) (Nat.cast_nonneg r)
(fun Z => htr _) j.2.1 j.2.2 N
obtain ⟨a,P,hP,hbound⟩ := hmain (3*s+1) hH (TripleVertices s)
(Equiv.Perm (GlobalVertices s q)) J T (tripleEquiv s) (fun _ => threeColor)
(fun _ c => triple_histogram_pos (by omega) c) (fun _ => M)
(fun j => liftLocal j.1) g hg (r:ℝ) hgb
refine ⟨a,fun i u v => P (i,u,v),fun i u v => hP (i,u,v),?_⟩
intro i u v t
let E (j : J) (t : T) :=
predicateAverage (fun N => crossVector threeColor N = crossVector threeColor (M t)) (g j a) -
P j (crossVector threeColor (M t))
have hjt : (E (i,u,v) t)^2 ≤ ∑ j, ∑ t, (E j t)^2 := by
exact (Finset.single_le_sum (fun z _ => sq_nonneg (E (i,u,v) z)) (Finset.mem_univ t)).trans
(Finset.single_le_sum (fun j _ => Finset.sum_nonneg (fun z _ => sq_nonneg (E j z)))
(Finset.mem_univ (i,u,v)))
have hb : (∑ j, ∑ t, (E j t)^2) ≤ η^2 := by
apply hbound.trans
have heq : (Fintype.card J : ℝ)*(Fintype.card T)*((2*(3*s+1:ℕ):ℝ)^(2*K))⁻¹*(r:ℝ)^2 =
(q:ℝ)*r^4*(Fintype.card (SampleIndex d))*((2*(3*s+1):ℝ)^(2*K))⁻¹ := by
simp only [J, T, Fintype.card_prod, Fintype.card_fin, Nat.cast_mul, Nat.cast_add, Nat.cast_ofNat]
ring
rw [heq]
exact hnum
have hab : |E (i,u,v) t| ≤ η := (sq_le_sq₀ (abs_nonneg _) hη).mp
(by simpa only [sq_abs] using hjt.trans hb)
have he : E (i,u,v) t = cellMatrix i (fun Z => B (Pairing.relabel a Z))
(samplePoint d (2*l) t) u v-P (i,u,v) (samplePoint d (2*l) t) := by
rw [cellMatrix_apply]
dsimp only [E, cellAverage, g]
rw [hM]
rwa [he] at hab
end MatchingInterpolation
end OAI
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