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f61e547 | 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 | 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
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