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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 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.Probability.FreeIsing.Posterior.SelectedPosteriorChannels
import OAI.Probability.FreeIsing.Trees.CutContinuationChannels
import OAI.Probability.FreeIsing.Trees.SelectedPathSegmentAttenuation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
open PathSideBranches CutContinuationChannels
def gaussianConeFactor {d : ℕ} (μ : Measure (TreeVertex d → IsingSpin))
(t : ℝ≥0) (theta : ℝ)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) (v : TreeVertex d) : ℝ :=
responseAttenuation theta (Real.artanh (selectedPosterior
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun s x => (t : ℝ) * spinSign s + x)
(Subtype.val : cone v → TreeVertex d) v z)) ^ 2
theorem gaussianConeFactor_mem_Icc {d : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) {theta : ℝ}
(hθ : theta ^ 2 ≤ 1) (z) (v : TreeVertex d) :
gaussianConeFactor μ t theta z v ∈ Set.Icc (0 : ℝ) 1 := by
unfold gaussianConeFactor
have hp := responseAttenuation_pos hθ (Real.artanh (selectedPosterior
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun s x => (t : ℝ) * spinSign s + x)
(Subtype.val : cone v → TreeVertex d) v z))
have hl := responseAttenuation_le_one hθ (Real.artanh (selectedPosterior
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun s x => (t : ℝ) * spinSign s + x)
(Subtype.val : cone v → TreeVertex d) v z))
exact ⟨sq_nonneg _, by nlinarith⟩
theorem exists_local_gaussianConeFactor {d : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) {theta : ℝ}
(hθ : theta ^ 2 ≤ 1) :
∃ F : ∀ v : TreeVertex d, ConeSample ℝ v → ℝ,
(∀ v, Measurable (F v)) ∧ (∀ v z, 0 ≤ F v z ∧ F v z ≤ 1) ∧
∀ z v, globalFactors F z v = gaussianConeFactor μ t theta z v := by
classical
have hex (v : TreeVertex d) : ∃ f : ConeSample ℝ v → ℝ,
Measurable f ∧ (∀ z, f z ∈ Set.Icc (0 : ℝ) 1) ∧
∀ z, gaussianConeFactor μ t theta z v = f (globalCone v z) := by
have hF : Measurable (Function.uncurry
(fun s x => (t : ℝ) * spinSign s + x)) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
exact exists_selectedAttenuation_local_channel
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) hF
(Subtype.val : cone v → TreeVertex d) id v hθ
choose f hf hb heq using hex
exact ⟨f, hf, hb, fun z v => (heq v z).symm⟩
theorem cutContinuationProduct_gaussianConeFactor {d k L : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) (theta : ℝ)
(p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) (z) :
cutContinuationProduct (gaussianConeFactor μ t theta z) p w =
∏ i ∈ Finset.Ico (k + 1) (k + 1 + L),
responseAttenuation theta
(forwardGaussianField μ t (appendCutContinuation p w).1 i z) ^ 2 := by
rw [Finset.prod_Ico_eq_prod_range, Nat.add_sub_cancel_left]
apply Finset.prod_congr rfl
intro j hj
simp only [gaussianConeFactor, forwardGaussianField, rootPathVertex,
ConnectedHull.takeVertex, Nat.add_assoc]
rfl
theorem integral_cutContinuationProduct_gaussianConeFactor_le {d k L : ℕ}
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (hd : 3 ≤ d)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t)
(p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) :
(∫ z, cutContinuationProduct (gaussianConeFactor μ t (Real.tanh beta) z) p w
∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) ≤
Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) *
min (Real.sqrt (t : ℝ)) 1) * (L : ℝ)) := by
have h := integral_forwardGaussianField_subset_attenuation_le_exp_critical
hμ hbeta hd hcritical ht (appendCutContinuation p w).1
(Finset.Ico (k + 1) (k + 1 + L)) (by
intro i hi
rw [(appendCutContinuation p w).2]
exact (Finset.mem_Ico.mp hi).2)
simpa only [cutContinuationProduct_gaussianConeFactor, Nat.card_Ico,
Nat.add_sub_cancel_left] using h
theorem measure_gaussian_restricted_path_average_ge_le_double_exp {d : ℕ}
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (hd : 3 ≤ d)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t)
(forbidden : Finset (Fin d)) (hforbidden : forbidden.card ≤ 1) (k L : ℕ)
{η ℓ : ℝ} (hη0 : 0 ≤ η) (hℓ : 0 ≤ ℓ)
(hη : η ≤ ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) / 4)
(hηlog : η ≤ Real.log ((d : ℝ) - 1) / 8)
(hlevel : ℓ / 2 - 1 ≤ (k + 1 : ℕ)) (htail : ℓ / 2 ≤ (L : ℝ))
(hlength : 8 * Real.log 2 ≤
ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) *
(min (Real.sqrt (t : ℝ)) 1 * ℓ)) :
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
{z | Real.exp (-(η * min (Real.sqrt (t : ℝ)) 1) * ℓ) ≤
(∑ v ∈ cutDescendants (avoidingTreeCut forbidden k) L,
rootPathProduct (gaussianConeFactor μ t (Real.tanh beta) z) v.1) /
((cutDescendants (avoidingTreeCut forbidden k) L).card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(((d : ℝ) - 1) ^ (ℓ / 4)) /
(2 * ((d : ℝ) - 1)))) := by
let := hμ.isProbabilityMeasure
have hθ0 : 0 < Real.tanh beta := by
have hn := tanh_nonneg_of_beta_nonneg hbeta
by_contra h
have hz : Real.tanh beta = 0 := le_antisymm (not_lt.mp h) hn
simp [hz] at hcritical
have hθ1 := Real.tanh_lt_one beta
have hθ : Real.tanh beta ^ 2 ≤ 1 := by nlinarith
have hc := (ComponentSignalAttenuation.lossCoefficient_pos hθ0 hθ1).le
have hω : 0 ≤ min (Real.sqrt (t : ℝ)) 1 :=
le_min (Real.sqrt_nonneg _) (by norm_num)
obtain ⟨F, hF, hb, heq⟩ := exists_local_gaussianConeFactor μ t hθ
have hfactor : globalFactors F = gaussianConeFactor μ t (Real.tanh beta) :=
funext (fun z => funext (fun v => heq z v))
have hmean (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) :
(∫ z, cutContinuationProduct (globalFactors F z) p w
∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => gaussianReal 0 t))) ≤
Real.exp (-(η * min (Real.sqrt (t : ℝ)) 1) * ℓ) / 4 := by
rw [hfactor]
change (∫ z, cutContinuationProduct (gaussianConeFactor μ t (Real.tanh beta) z) p w
∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) ≤ _
refine (integral_cutContinuationProduct_gaussianConeFactor_le
hμ hbeta hd hcritical ht p w).trans ?_
calc
_ ≤ Real.exp (-ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) *
(min (Real.sqrt (t : ℝ)) 1 * ℓ) / 2) := by
apply Real.exp_le_exp.mpr
have hm := mul_le_mul_of_nonneg_left htail (mul_nonneg hc hω)
nlinarith
_ ≤ Real.exp (-η * (min (Real.sqrt (t : ℝ)) 1 * ℓ)) / 4 :=
attenuation_le_quarter_threshold (mul_nonneg hω hℓ) hη hlength
_ = _ := by ring_nf
have hq : η * min (Real.sqrt (t : ℝ)) 1 ≤ Real.log ((d : ℝ) - 1) / 8 :=
(mul_le_of_le_one_right hη0 (min_le_right _ _)).trans hηlog
have h := measure_restricted_path_average_ge_le_double_exp hμ
(by linarith : -1 ≤ Real.tanh beta) hθ1.le (by omega : 2 ≤ d)
(fun _ : TreeVertex d => gaussianReal 0 t) F hF hb forbidden hforbidden k L
hℓ hq hlevel (fun p _ w => hmean p w)
simpa only [hfactor, PlantedSymmetry.gaussianNoiseLaw] using h
end Problem367
end
end OAI
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