| import OAI.Probability.FreeIsing.Conditioning.CenteredCopyDifference |
| import OAI.Probability.FreeIsing.Spectral.NoisyLawSpectral |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| namespace Problem367.CopyDifference |
| open PlantedStationarity |
|
|
| variable {d : ℕ} {A B : Type*} [MeasurableSpace A] [MeasurableSpace B] |
| variable {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| variable (hμ : HasFreeIsingLaw theta μ) (ν : Measure A) [IsProbabilityMeasure ν] |
| variable {O : IsingSpin → A → B} (hO : Measurable (Function.uncurry O)) |
| variable {π : Measure ((TreeVertex d → B) × (TreeVertex d → B))} |
| variable (hp : MeasurePreserving Prod.fst π (NoisyLawSpectral.law μ ν O)) |
| variable (hq : MeasurePreserving Prod.snd π (NoisyLawSpectral.law μ ν O)) |
| variable (hπ : ∀ g : TreeAut d, MeasurePreserving |
| (Prod.map (relabel g) (relabel g)) π π) |
|
|
| def noisyAverageOnDifferences : |
| rangeClosure (operator hp hq) →L[ℝ] rangeClosure (operator hp hq) := |
| averageOnDifferences hp hq (TreeLawKoopman.reflection (A := B)) |
| (fun c => Prod.map (relabel (prefixTreeAut c)) (relabel (prefixTreeAut c))) |
| (fun c => measurePreserving_coordinateMap_relabel hμ ν O hO (prefixTreeAut c)) |
| (fun c => hπ (prefixTreeAut c)) (fun _ => rfl) (fun _ => rfl) |
|
|
| omit [MeasurableSpace B] in |
| private theorem diagonalReflection_involutive (c : Fin d) : |
| Function.Involutive |
| (Prod.map (TreeLawKoopman.reflection (A := B) c) |
| (TreeLawKoopman.reflection (A := B) c)) := by |
| intro p |
| exact Prod.ext (TreeLawKoopman.reflection_involutive c p.1) |
| (TreeLawKoopman.reflection_involutive c p.2) |
|
|
| theorem noisyAverageOnDifferences_isSelfAdjoint : |
| IsSelfAdjoint (noisyAverageOnDifferences hμ ν hO hp hq hπ) := |
| averageOnDifferences_isSelfAdjoint hp hq _ _ _ _ diagonalReflection_involutive |
| (fun _ => rfl) (fun _ => rfl) |
|
|
| theorem noisyAverageOnDifferences_norm_le (hd : 3 ≤ d) |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) : |
| ‖noisyAverageOnDifferences hμ ν hO hp hq hπ‖ ≤ |
| 2 * Real.sqrt ((d : ℝ) - 1) / d := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let : IsProbabilityMeasure (NoisyLawSpectral.law μ ν O) := |
| (Measure.isProbabilityMeasure_map_iff |
| (measurable_coordinateMap hO).aemeasurable).mpr inferInstance |
| apply norm_averageOnDifferences_le hp hq _ _ _ _ |
| TreeLawKoopman.reflection_involutive diagonalReflection_involutive |
| (fun _ => rfl) (fun _ => rfl) |
| · positivity |
| · exact NoisyLawSpectral.centeredAverage_norm_le_critical hμ hd h0 h1 hthreshold ν hO |
|
|
| end Problem367.CopyDifference |
|
|
| end |
|
|
| end OAI |
|
|
