| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.OwnTotalEnergyNo |
| import OAI.Computability.VertexCover.Analysis.Poincare |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| namespace VertexCover.LabelCover |
| open MeasureTheory ProbabilityTheory |
|
|
| theorem ownMean_poincare (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (j : J) (i : Φ.Query d) : |
| variance (Φ.ownMean J frozen c0 A hA j i) |
| (VertexCover.Cube.law (Fin (Φ.WeightDimension d))) ≤ |
| 2 * ∫ s, ∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2 |
| ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d)) := by |
| apply VertexCover.Cube.poincare (Φ.ownMean_lipschitz J frozen c0 A hA j i) |
| (Φ.ownGradient J frozen c0 A hA j i) |
| (Φ.ownGradient_measurable J frozen c0 A hA j i) |
| · intro s k |
| exact VertexCover.Threshold.coordinate_bound (Φ.ownGradient_block_bound J frozen c0 A hA j i s) k |
| · exact Φ.ownMean_partial_ae J frozen c0 A hA j i |
|
|
| noncomputable def ownTotalVariance (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : ℝ := |
| ∑ j : J, VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J => |
| variance (Φ.ownMean J frozen c0 A hA j (Φ.query (Φ.spliceSeeds J frozen hidden) j)) |
| (VertexCover.Cube.law (Fin (Φ.WeightDimension d)))) |
|
|
| theorem ownGradient_sq_integrable (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (j : J) (i : Φ.Query d) : |
| Integrable (fun s => ∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2) |
| (VertexCover.Cube.law (Fin (Φ.WeightDimension d))) := by |
| apply integrable_finsetSum |
| intro k hk |
| apply Integrable.of_bound ((Φ.ownGradient_measurable J frozen c0 A hA j i k).pow_const 2).aestronglyMeasurable 1 |
| filter_upwards [] with s |
| have hb := VertexCover.Threshold.coordinate_bound (Φ.ownGradient_block_bound J frozen c0 A hA j i s) k |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| nlinarith [abs_nonneg (Φ.ownGradient J frozen c0 A hA j i s k), |
| sq_abs (Φ.ownGradient J frozen c0 A hA j i s k)] |
|
|
| theorem ownTotalVariance_le_energy (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : |
| Φ.ownTotalVariance J frozen c0 A hA ≤ 2 * Φ.ownTotalEnergy J frozen c0 A hA := by |
| classical |
| unfold ownTotalVariance ownTotalEnergy |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro j hj |
| rw [VertexCover.integral_finiteMean _ _ (fun hidden => |
| Φ.ownGradient_sq_integrable J frozen c0 A hA j (Φ.query (Φ.spliceSeeds J frozen hidden) j))] |
| calc |
| _ ≤ VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J => |
| 2 * ∫ s, ∑ k, (Φ.ownGradient J frozen c0 A hA j |
| (Φ.query (Φ.spliceSeeds J frozen hidden) j) s k)^2 |
| ∂VertexCover.Cube.law (Fin (Φ.WeightDimension d))) := |
| VertexCover.finiteMean_mono (fun hidden => Φ.ownMean_poincare J frozen c0 A hA j _) |
| _ = _ := by simp only [VertexCover.finiteMean, ← Finset.mul_sum]; ring |
| |
| theorem ownTotalVariance_no_upper (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m) |
| (hval : Φ.value ≤ Parameters.σ m) |
| (J : Finset (Fin (Parameters.d m))) (hJ : J.card = Parameters.h m) |
| (frozen : Φ.Seeds (Parameters.d m)) (c0 : Φ.Coordinate (Parameters.d m) → ℝ) |
| (A : Finset (Φ.Coordinate (Parameters.d m) → ℝ)) (hA : A.Nonempty) : |
| Φ.ownTotalVariance J frozen c0 A hA < 7 := by |
| have he := Φ.ownTotalVariance_le_energy J frozen c0 A hA |
| have hu := Φ.ownTotalEnergy_no_upper m hm hval J hJ frozen c0 A hA |
| linarith |
|
|
| end VertexCover.LabelCover |
|
|
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
