| 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 |
|
|
