| import OAI.Probability.FreeIsing.Conditioning.CenteredCopyDifference |
| import OAI.Probability.FreeIsing.Gaussian.BernoulliSpectral |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
|
|
| namespace Problem367.CopyDifference |
|
|
| variable {d : ℕ} {B : Type*} [MeasurableSpace B] {ν : Measure B} |
| variable {p q : B → (TreeVertex d → ℝ)} |
|
|
| def bernoulliAverageOnDifferences |
| (hp : MeasurePreserving p ν (labelMeasure d)) |
| (hq : MeasurePreserving q ν (labelMeasure d)) |
| (σ : Fin d → B → B) (hσ : ∀ c, MeasurePreserving (σ c) ν ν) |
| (hpc : ∀ c, p ∘ σ c = TreeKoopman.reflection c ∘ p) |
| (hqc : ∀ c, q ∘ σ c = TreeKoopman.reflection c ∘ q) : |
| rangeClosure (operator hp hq) →L[ℝ] rangeClosure (operator hp hq) := |
| averageOnDifferences hp hq TreeKoopman.reflection σ |
| TreeKoopman.reflection_measurePreserving hσ hpc hqc |
|
|
| theorem bernoulliAverageOnDifferences_isSelfAdjoint |
| (hp : MeasurePreserving p ν (labelMeasure d)) |
| (hq : MeasurePreserving q ν (labelMeasure d)) |
| (σ : Fin d → B → B) (hσ : ∀ c, MeasurePreserving (σ c) ν ν) |
| (hσinv : ∀ c, Function.Involutive (σ c)) |
| (hpc : ∀ c, p ∘ σ c = TreeKoopman.reflection c ∘ p) |
| (hqc : ∀ c, q ∘ σ c = TreeKoopman.reflection c ∘ q) : |
| IsSelfAdjoint (bernoulliAverageOnDifferences hp hq σ hσ hpc hqc) := |
| averageOnDifferences_isSelfAdjoint hp hq TreeKoopman.reflection σ |
| TreeKoopman.reflection_measurePreserving hσ hσinv hpc hqc |
|
|
| theorem bernoulliAverageOnDifferences_norm_le (hd : 3 ≤ d) |
| (hp : MeasurePreserving p ν (labelMeasure d)) |
| (hq : MeasurePreserving q ν (labelMeasure d)) |
| (σ : Fin d → B → B) (hσ : ∀ c, MeasurePreserving (σ c) ν ν) |
| (hσinv : ∀ c, Function.Involutive (σ c)) |
| (hpc : ∀ c, p ∘ σ c = TreeKoopman.reflection c ∘ p) |
| (hqc : ∀ c, q ∘ σ c = TreeKoopman.reflection c ∘ q) : |
| ‖bernoulliAverageOnDifferences hp hq σ hσ hpc hqc‖ ≤ |
| 2 * Real.sqrt ((d : ℝ) - 1) / d := by |
| apply norm_averageOnDifferences_le hp hq TreeKoopman.reflection σ |
| TreeKoopman.reflection_measurePreserving hσ |
| TreeKoopman.reflection_involutive hσinv hpc hqc |
| · positivity |
| · exact TreeKoopman.centeredAverage_norm_le hd |
|
|
| end Problem367.CopyDifference |
|
|
| end |
|
|
| end OAI |
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|
