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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Fields.FieldSpinMeasurability
namespace OAI
/-! Preservation of the common-level law by the published finite-field
terminal tilt, including the independent scalar-field spins. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
lemma field_leaf_depth_pair_conditioning {N n : ℕ}
(ν : Measure (LabeledLeaf n)) [IsProbabilityMeasure ν]
(Y : LabeledLeaf n → Fin N → ℝ)
(he : Integrable (fun p : Spin N × LabeledLeaf n => Real.exp (fieldEnergy (Y p.2) p.1))
((uniformSpinPrior N : Measure (Spin N)).prod ν))
(ψ : ℕ → ℝ) {C : ℝ} (hψ : ∀ d, |ψ d| ≤ C) :
referenceReplicaMean ((uniformSpinPrior N : Measure (Spin N)).prod ν)
(fun p => fieldEnergy (Y p.2) p.1)
(fun σ : Fin 2 → Spin N × LabeledLeaf n => ψ (labeledCommonDepth n (σ 0).2 (σ 1).2)) =
referenceReplicaMean ν (fun α => ∑ i, Real.log (Real.cosh (Y α i)))
(fun α : Fin 2 → LabeledLeaf n => ψ (labeledCommonDepth n (α 0) (α 1))) := by
let H := fun p : Spin N × LabeledLeaf n => fieldEnergy (Y p.2) p.1
have ht : spinLeafTerminal H = (fun α => ∑ i, Real.log (Real.cosh (Y α i))) := by
funext α
change finiteLogIntegral (uniformSpinPrior N : Measure (Spin N)) (fieldEnergy (Y α)) = _
rw [finiteLogIntegral_uniformSpinPrior, logPartition_fieldEnergy]
rw [spinLeafPair_conditioning ν H he _ ⟨C, fun σ => hψ _⟩, ht]
congr 1
funext α
change (∫ _ : Fin 2 → Spin N, ψ (labeledCommonDepth n (α 0) (α 1)) ∂_) = _
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul]
lemma field_published_depth_test (hpub : PanchenkoTalagrandFieldPairInput)
(N : ℕ) (hN : 0 < N) (h : FieldStep) (ψ : ℕ → ℝ)
{C : ℝ} (hψ : ∀ d, |ψ d| ≤ C) (z : Fin N → ℝ) :
fieldVectorTiltedPairMean N h z (fun p => ψ p.1) =
∫ s, ψ (fieldLevelIndex h s).val ∂pathMeasure := by
have hm : Measurable (fun p : ℕ × ((Fin N → ℝ) × (Fin N → ℝ)) => ψ p.1) :=
(measurable_of_countable ψ).comp measurable_fst
rw [hpub N hN h z (fun p => ψ p.1) hm ⟨C, fun p => hψ p.1⟩]
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul]
exact fieldCommonLevel_integral h (fun i => ψ i.val)
def fieldDepthPairIntegrand (N : ℕ) (h : FieldStep) (z : Fin N → ℝ)
(p : LabeledTree h.depth × (ForestVertex h.depth → Fin N → ℝ)) (ψ : ℕ → ℝ) : ℝ :=
referenceReplicaMean
((uniformSpinPrior N : Measure (Spin N)).prod (labeledLeafLaw h.depth p.1))
(fun s : Spin N × LabeledLeaf h.depth => fieldEnergy (fieldVectorEndpoint N h p z s.2) s.1)
(fun σ : Fin 2 → Spin N × LabeledLeaf h.depth =>
ψ (labeledCommonDepth h.depth (σ 0).2 (σ 1).2))
def fieldVectorDepthPairMean (N : ℕ) (h : FieldStep) (z : Fin N → ℝ) (ψ : ℕ → ℝ) : ℝ :=
∫ p, fieldDepthPairIntegrand N h z p ψ ∂fieldVectorCoordinateLaw N h
lemma measurable_fieldDepthPairIntegrand (N : ℕ) (h : FieldStep) (ψ : ℕ → ℝ) :
Measurable (fun p : (Fin N → ℝ) ×
(LabeledTree h.depth × (ForestVertex h.depth → Fin N → ℝ)) =>
fieldDepthPairIntegrand N h p.1 p.2 ψ) := by
let D : (Fin 2 → Spin N × LabeledLeaf h.depth) → ℝ := fun σ =>
ψ (labeledCommonDepth h.depth (σ 0).2 (σ 1).2)
have hm := measurable_cascadeCoordinateReplicaMean h.depth
(uniformSpinPrior N : Measure (Spin N)) (measurable_fieldEnergy_map N) D
have hc : Measurable (fun p : (Fin N → ℝ) ×
(LabeledTree h.depth × (ForestVertex h.depth → Fin N → ℝ)) =>
(p.1, fieldEnergyCoordinates N h p.2)) :=
measurable_fst.prodMk ((measurable_fieldEnergyCoordinates N h).comp measurable_snd)
have he (z : Fin N → ℝ)
(p : LabeledTree h.depth × (ForestVertex h.depth → Fin N → ℝ)) :
cascadeCoordinateEnergy h.depth (fieldEnergy z) (fieldEnergyCoordinates N h p) =
fun s => fieldEnergy (fieldVectorEndpoint N h p z s.2) s.1 :=
funext (fieldEnergyCoordinates_energy N h p z)
have hh := hm.comp hc
simp only [Function.comp_def] at hh
simp_rw [he] at hh
simpa only [labeledSpinReference, fieldEnergyCoordinates, fieldDepthPairIntegrand, D] using hh
lemma measurable_fieldVectorDepthPairMean (N : ℕ) (h : FieldStep) (ψ : ℕ → ℝ) :
Measurable (fun z => fieldVectorDepthPairMean N h z ψ) :=
(measurable_fieldDepthPairIntegrand N h ψ).stronglyMeasurable.integral_prod_right'.measurable
theorem field_vector_depth_pair_evaluation (hpub : PanchenkoTalagrandFieldPairInput)
(N : ℕ) (hN : 0 < N) (h : FieldStep) (ψ : ℕ → ℝ)
{C : ℝ} (hψ : ∀ d, |ψ d| ≤ C) (z : Fin N → ℝ) :
fieldVectorDepthPairMean N h z ψ = ∫ s, ψ (fieldLevelIndex h s).val ∂pathMeasure := by
rw [← field_published_depth_test hpub N hN h ψ hψ z]
unfold fieldVectorDepthPairMean fieldVectorTiltedPairMean
apply integral_congr_ae
filter_upwards [field_vector_spin_exp_integrable N hN h z] with p hp
exact field_leaf_depth_pair_conditioning (labeledLeafLaw h.depth p.1)
(fieldVectorEndpoint N h p z) hp ψ hψ
end InvariantIsing
end
end OAI
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