| import OAI.Combinatorics.MatchingPSD.MatrixInterpolation |
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| namespace OAI |
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| namespace MatchingInterpolation |
| open MatchingSmoothing PerfectMatchingPSD Matrix |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
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| abbrev SampleIndex (d : ℕ) := Fin 5 ⊕ GridIndex d |
| noncomputable def samplePoint (d : ℕ) (L : ℝ) : SampleIndex d → Point := |
| Sum.elim specialPoint (scaledNode L) |
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| 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] |
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| 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 |
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| 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 |
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| end MatchingInterpolation |
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| end OAI |
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