| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.PrivateProjectionSurvival |
| import OAI.Computability.VertexCover.Reduction.BatchesTripleSum |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
| open VertexCover.Restriction |
| open EnergyForm.Projection |
| open scoped BigOperators |
|
|
| theorem weightGroup_residual_budget (Φ : LabelCover) {d : ℕ} |
| (Q : (Φ.restrictionForm d).Projection) |
| (hQ : ∀ k f, Φ.weightProjection k (Q f) = Q (Φ.weightProjection k f)) |
| (f : Φ.RestrictionSpace d) : |
| (∑ j : Fin d, (Φ.restrictionForm d).energy (Q.residual (Φ.weightGroup j f))) ≤ |
| (Φ.restrictionForm d).energy (Q.residual f) := |
| finDifference_residual_budget _ Φ.weightProjection_commute Q hQ f |
|
|
| theorem weightGroup_pair_budget (Φ : LabelCover) {d : ℕ} |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : |
| (∑ j : Fin d, ∑ k : Fin d, |
| ((Φ.restrictionForm d).energy ((Φ.weightProjection k).residual |
| (Φ.weightGroup j (Φ.restrictionFunction A hA))) + |
| (Φ.restrictionForm d).energy ((Φ.starProjection k).residual |
| (Φ.weightGroup j (Φ.restrictionFunction A hA))))) ≤ 8*d := by |
| rw [Finset.sum_comm] |
| calc |
| _ ≤ ∑ k : Fin d, |
| ((Φ.restrictionForm d).energy ((Φ.weightProjection k).residual |
| (Φ.restrictionFunction A hA)) + |
| (Φ.restrictionForm d).energy ((Φ.starProjection k).residual |
| (Φ.restrictionFunction A hA))) := by |
| apply Finset.sum_le_sum |
| intro k _ |
| rw [Finset.sum_add_distrib] |
| apply add_le_add |
| · exact Φ.weightGroup_residual_budget (Φ.weightProjection k) |
| (fun i f => Φ.weightProjection_commute i k f) _ |
| · exact Φ.weightGroup_residual_budget (Φ.starProjection k) |
| (fun i f => (Φ.condition_weightProjection_commute (Φ.starKey k) i f).symm) _ |
| _ ≤ _ := by |
| rw [Finset.sum_add_distrib] |
| have hw := Φ.restriction_weight_budget A hA |
| have hs := Φ.restriction_star_budget A hA |
| linarith |
|
|
| theorem weightGroup_seed_budget (Φ : LabelCover) {d : ℕ} |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : |
| (∑ j : Fin d, ∑ e : PositionPair d, |
| (Φ.restrictionForm d).energy ((Φ.seedProjection e).residual |
| (Φ.weightGroup j (Φ.restrictionFunction A hA)))) ≤ 8*(d:ℝ)^2 := by |
| rw [Finset.sum_comm] |
| calc |
| _ ≤ ∑ e : PositionPair d, (Φ.restrictionForm d).energy |
| ((Φ.seedProjection e).residual (Φ.restrictionFunction A hA)) := by |
| apply Finset.sum_le_sum |
| intro e _ |
| exact Φ.weightGroup_residual_budget (Φ.seedProjection e) |
| (fun i f => (Φ.condition_weightProjection_commute |
| (fun seed : Φ.Seeds d => Function.update seed e ⟨0, Φ.M_pos⟩) i f).symm) _ |
| _ ≤ _ := Φ.restriction_seed_budget A hA |
|
|
| theorem privateProjection_batch_bound (Φ : LabelCover) {d h : ℕ} (hh : 3 ≤ h) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : |
| ((d-1).choose (h-1) : ℝ) * (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA) ≤ |
| (∑ J ∈ batches d h, ∑ j ∈ J, (Φ.restrictionForm d).energy |
| (Φ.privateProjection J j (Φ.restrictionFunction A hA))) + |
| ((d-2).choose (h-2) : ℝ)*(8*d) + ((d-3).choose (h-3) : ℝ)*(8*(d:ℝ)^2) := by |
| classical |
| let B := Φ.restrictionForm d |
| let f := Φ.restrictionFunction A hA |
| let pairLoss := fun j k : Fin d => B.energy ((Φ.weightProjection k).residual (Φ.weightGroup j f)) + |
| B.energy ((Φ.starProjection k).residual (Φ.weightGroup j f)) |
| let seedLoss := fun (j : Fin d) (e : PositionPair d) => |
| B.energy ((Φ.seedProjection e).residual (Φ.weightGroup j f)) |
| have hsurv : (∑ J ∈ batches d h, ∑ j ∈ J, B.energy (Φ.weightGroup j f)) ≤ |
| (∑ J ∈ batches d h, ∑ j ∈ J, B.energy (Φ.privateProjection J j f)) + |
| (∑ J ∈ batches d h, ∑ j ∈ J, ∑ k ∈ J.erase j, pairLoss j k) + |
| (∑ J ∈ batches d h, ∑ j ∈ J, |
| ∑ e ∈ (internalPairs J).filter (fun e => ¬ (e.1.1 = j ∨ e.1.2 = j)), seedLoss j e) := by |
| simp only [← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum |
| intro J _ |
| apply Finset.sum_le_sum |
| intro j _ |
| exact Φ.privateProjection_survival J j f |
| have hpair := batches_pair_sum_le (by omega : 2 ≤ h) pairLoss |
| (fun j k => add_nonneg (B.energy_nonneg _) (B.energy_nonneg _)) |
| have htriple := batches_triple_sum_le hh |
| (fun e : PositionPair d => e.1.1) (fun e : PositionPair d => e.1.2) |
| (fun e => ne_of_lt e.2) seedLoss (fun _ _ => B.energy_nonneg _) |
| have hsets (J : Finset (Fin d)) (j : Fin d) : |
| (internalPairs J).filter (fun e => ¬ (e.1.1 = j ∨ e.1.2 = j)) = |
| Finset.univ.filter (fun e : PositionPair d => |
| e.1.1 ∈ J ∧ e.1.2 ∈ J ∧ ¬ (e.1.1 = j ∨ e.1.2 = j)) := by |
| ext e |
| simp only [internalPairs, Finset.mem_filter, Finset.mem_univ, true_and] |
| tauto |
| simp_rw [hsets] at hsurv |
| rw [batches_singleton_sum (by omega : 1 ≤ h), Φ.weightGroup_energy_sum A hA] at hsurv |
| have hpb : (∑ j, ∑ k, pairLoss j k) ≤ 8*d := Φ.weightGroup_pair_budget A hA |
| have hsb : (∑ j, ∑ e, seedLoss j e) ≤ 8*(d:ℝ)^2 := Φ.weightGroup_seed_budget A hA |
| have hpb' := mul_le_mul_of_nonneg_left hpb (Nat.cast_nonneg ((d-2).choose (h-2)) : (0:ℝ) ≤ _) |
| have hsb' := mul_le_mul_of_nonneg_left hsb (Nat.cast_nonneg ((d-3).choose (h-3)) : (0:ℝ) ≤ _) |
| dsimp only [B, f] at hsurv |
| linarith |
|
|
| end VertexCover.LabelCover |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
| end |
| end |
|
|
| end OAI |
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