File size: 3,920 Bytes
b317718 | 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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import OAI.Probability.FreeIsing.Ising.ObservationLaw
import OAI.Probability.FreeIsing.Trees.NoisyEdgeDecay
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.IsingObservationInnovation
open PlantedStationarity CopyDifference ColoredEdgeLifts
def uniformObservationChannel (s : IsingSpin) (u : ℝ) : WienerSampler.Path :=
⟨fun t => WienerSampler.sample u t + (t : ℝ) * spinSign s,
(WienerSampler.sample u).continuous.add
(NNReal.continuous_coe.mul continuous_const)⟩
theorem measurable_uniformObservationChannel :
Measurable (Function.uncurry uniformObservationChannel) := by
apply WienerSampler.measurable_path_of_eval
intro t
have hs : Measurable spinSign := by intro s hs; trivial
exact ((continuous_eval_const t).measurable.comp
(WienerSampler.measurable_sample.comp measurable_snd)).add
((hs.comp measurable_fst).const_mul (t : ℝ))
theorem observationLaw_eq_noisyLaw {d : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) :
observationLaw μ = NoisyLawSpectral.law μ uniform01 uniformObservationChannel := rfl
variable {d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
variable (hμ : HasFreeIsingLaw theta μ)
theorem observationLaw_centeredAverage_norm_le_critical [IsProbabilityMeasure μ]
(hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) :
‖TreeLawKoopman.centeredAverage (observationLaw μ)
(measurePreserving_observationLaw_relabel hμ)‖ ≤
2 * Real.sqrt ((d : ℝ) - 1) / d :=
NoisyLawSpectral.centeredAverage_norm_le_critical hμ hd h0 h1 hthreshold
uniform01 measurable_uniformObservationChannel
variable {π : Measure (Paths d × Paths d)}
variable (hp : MeasurePreserving Prod.fst π (observationLaw μ))
variable (hq : MeasurePreserving Prod.snd π (observationLaw μ))
variable (hπ : ∀ g : TreeAut d, MeasurePreserving
(Prod.map (relabel g) (relabel g)) π π)
def observationDifferenceReflection (c : Fin d) :
rangeClosure (operator hp hq) ≃ₗᵢ[ℝ] rangeClosure (operator hp hq) :=
noisyDifferenceReflection hμ uniform01 measurable_uniformObservationChannel hp hq hπ c
theorem observationLaw_differenceAverage_norm_le_critical
(hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) :
‖neighborAverage (observationDifferenceReflection hμ hp hq hπ)‖ ≤
2 * Real.sqrt ((d : ℝ) - 1) / d := by
change ‖neighborAverage (noisyDifferenceReflection hμ uniform01
measurable_uniformObservationChannel hp hq hπ)‖ ≤ _
rw [neighborAverage_noisyDifferenceReflection hμ uniform01
measurable_uniformObservationChannel hp hq hπ]
exact noisyAverageOnDifferences_norm_le hμ uniform01
measurable_uniformObservationChannel hp hq hπ hd h0 h1 hthreshold
theorem observationLaw_forwardOperator_pow_bound
(hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) :
∃ K : ℝ, 0 ≤ K ∧ ∀ n : ℕ,
‖forwardOperator (by omega : 0 < d)
(observationDifferenceReflection hμ hp hq hπ) ((d : ℝ) - 1)⁻¹ ^ n‖ ≤
K * ((n : ℝ) + 1) * (Real.sqrt ((d : ℝ) - 1))⁻¹ ^ n :=
noisy_forwardOperator_pow_bound hμ uniform01 measurable_uniformObservationChannel
hp hq hπ hd h0 h1 hthreshold
theorem observationLaw_summable_forwardOperator_pow
(hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) :
Summable (fun n : ℕ => ‖forwardOperator (by omega : 0 < d)
(observationDifferenceReflection hμ hp hq hπ) ((d : ℝ) - 1)⁻¹ ^ n‖) :=
summable_noisy_forwardOperator_pow hμ uniform01 measurable_uniformObservationChannel
hp hq hπ hd h0 h1 hthreshold
end Problem367.IsingObservationInnovation
end
end OAI
|