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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 | 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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