| import OAI.Probability.FreeIsing.Energy.RelabeledComponentAttenuation |
| import OAI.Probability.FreeIsing.Trees.NoisyPathLocality |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367.PathSideBranches |
|
|
| theorem range_relabel_componentIndexVertex_eq_cone |
| {d : ℕ} (g : TreeAut d) (r : TreeVertex d) (c : Fin d) |
| (hgeo : ∀ u : TreeVertex d, u ∈ cone r ↔ [c] <+: (g.toEquiv u).1) : |
| Set.range (fun i => g.toEquiv.symm (componentIndexVertex (c := c) i)) = |
| Set.range (Subtype.val : cone r → TreeVertex d) := by |
| rw [Subtype.range_coe_subtype] |
| ext u |
| constructor |
| · rintro ⟨i, rfl⟩ |
| apply (hgeo _).mpr |
| simp only [Equiv.apply_symm_apply] |
| change [c] <+: c :: i.2.1 |
| simp |
| · intro hu |
| have hp := (hgeo u).mp hu |
| have hm : g.toEquiv u ∈ Set.range (componentIndexVertex (c := c)) := by |
| apply mem_range_componentIndexVertex.mpr |
| obtain ⟨w, hw⟩ := hp |
| exact ⟨w, by simpa using hw.symm⟩ |
| obtain ⟨i, hi⟩ := hm |
| refine ⟨i, ?_⟩ |
| change g.toEquiv.symm (componentIndexVertex i) = u |
| rw [hi, Equiv.symm_apply_apply] |
|
|
| end Problem367.PathSideBranches |
|
|
| namespace Problem367.ComponentSignalAttenuation |
|
|
| open PlantedSymmetry PathSideBranches |
|
|
| theorem condExp_attenuation_le_critical_cone_complement |
| {d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw theta μ) (hθ0 : 0 < theta) (hθ1 : theta < 1) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (r : TreeVertex d) (hr : r.1 ≠ []) |
| {B : Type*} [mB : MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| (G : (({u : TreeVertex d // u ∉ cone r} → IsingSpin) × |
| ({u : TreeVertex d // u ∉ cone r} → ℝ)) → B) (hG : Measurable G) |
| (shift : B → ℝ) (hs : Measurable shift) : |
| let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t) |
| let X := G ∘ noisyRegionArray (cone r)ᶜ |
| let Q := selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z) |
| (Subtype.val : cone r → TreeVertex d) r |
| ∀ᵐ w ∂P, |
| (P[fun v => responseAttenuation theta |
| (shift (X v) + statisticMessage theta (Q v)) ^ 2 | mB.comap X]) w ≤ |
| 1 - lossCoefficient theta * min (Real.sqrt (t : ℝ)) 1 := by |
| obtain ⟨g, c, hp, hc, hgeo⟩ := exists_reroot_cone r hr |
| have hroot : g.toEquiv.symm (RootedBroadcast.branchRoot c) = r := by |
| apply g.toEquiv.injective |
| rw [Equiv.apply_symm_apply, hc] |
| rfl |
| have hchannelField := selectedObservationField_eq_of_range_eq |
| (fun s z => (t : ℝ) * spinSign s + z) |
| (range_relabel_componentIndexVertex_eq_cone g r c hgeo) |
| have hX : ((G ∘ decodeConeComplement (A := ℝ) g r c hgeo) ∘ noisyRootOtherArray c) ∘ |
| Prod.map (relabel g) (relabel g) = G ∘ noisyRegionArray (cone r)ᶜ := by |
| funext z |
| exact congrArg G (decodeConeComplement_relabel g r c hgeo z) |
| have h := condExp_attenuation_le_critical_relabel_root_others |
| hμ hθ0 hθ1 c hcritical ht g (G ∘ decodeConeComplement g r c hgeo) |
| (hG.comp (measurable_decodeConeComplement g r c hgeo)) shift hs |
| dsimp only at h ⊢ |
| simpa only [hX, hroot, selectedPosterior, hchannelField] using h |
|
|
| theorem condExp_attenuation_le_critical_cone_of_measurable |
| {d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw theta μ) (hθ0 : 0 < theta) (hθ1 : theta < 1) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (r : TreeVertex d) (hr : r.1 ≠ []) |
| {B : Type*} [mB : MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| (X : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) → B) |
| (hX : Measurable[noisyRegionSigma (A := ℝ) (cone r)ᶜ] X) |
| (shift : B → ℝ) (hs : Measurable shift) : |
| let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t) |
| let Q := selectedPosterior P (fun s z => (t : ℝ) * spinSign s + z) |
| (Subtype.val : cone r → TreeVertex d) r |
| ∀ᵐ w ∂P, |
| (P[fun v => responseAttenuation theta |
| (shift (X v) + statisticMessage theta (Q v)) ^ 2 | mB.comap X]) w ≤ |
| 1 - lossCoefficient theta * min (Real.sqrt (t : ℝ)) 1 := by |
| obtain ⟨G, hG, rfl⟩ := hX.exists_eq_measurable_comp |
| exact condExp_attenuation_le_critical_cone_complement |
| hμ hθ0 hθ1 hcritical ht r hr G hG shift hs |
|
|
| end Problem367.ComponentSignalAttenuation |
|
|
| end |
|
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| end OAI |
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|
