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a2ee19a | 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 Mathlib
import OAI.Probability.Perceptron.Variational.LabelPairTestValue
namespace OAI
noncomputable section
namespace SphericalPerceptronFreeEnergy
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology BoundedContinuousFunction NNReal ENNReal BigOperators
abbrev WeightedRestorationRange (f : ℝ →ᵇ ℝ) (C : ℝ) := Icc (-C) C × RestorationRange f
def compactWeightedLaw {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (f : ℝ →ᵇ ℝ) (C : ℝ)
(A : Ω→ℝ) (hA : Measurable A) (Z : Ω→RestorationRange f) (hZ : Measurable Z)
(hb : ∀ x, |A x|≤C) : ProbabilityMeasure (WeightedRestorationRange f C) :=
⟨μ.map (fun x => (⟨A x,abs_le.mp (hb x)⟩,Z x)),
(Measure.isProbabilityMeasure_map_iff ((hA.subtype_mk).prodMk hZ).aemeasurable).2 inferInstance⟩
lemma compactWeightedLaw_integral {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (f : ℝ →ᵇ ℝ) (C : ℝ)
(A : Ω→ℝ) (hA : Measurable A) (Z : Ω→RestorationRange f) (hZ : Measurable Z)
(hb : ∀ x, |A x|≤C) (J : WeightedRestorationRange f C → ℝ) (hJ : Measurable J) :
(∫ x, J x ∂(compactWeightedLaw μ f C A hA Z hZ hb : Measure _))=
∫ x, J (⟨A x,abs_le.mp (hb x)⟩,Z x) ∂μ :=
integral_map ((hA.subtype_mk).prodMk hZ).aemeasurable hJ.aestronglyMeasurable
lemma compactWeighted_witness {Ω : ℕ→Type*} [∀ n, MeasurableSpace (Ω n)]
(μ : (n : ℕ)→Measure (Ω n)) [∀ n, IsProbabilityMeasure (μ n)]
(f : ℝ →ᵇ ℝ) (C : ℝ) (A : (n : ℕ)→Ω n→ℝ) (Z : (n : ℕ)→Ω n→RestorationRange f)
(hA : ∀ n, Measurable (A n)) (hZ : ∀ n, Measurable (Z n)) (hb : ∀ n x, |A n x|≤C)
(b : ℕ→ℝ) (hm : ∀ j, Tendsto (fun n => ∫ x, A n x*(Z n x).val^j ∂μ n) atTop (𝓝 (b j)))
(J : C(RestorationRange f,ℝ)) (L : ℝ)
(hL : Tendsto (fun n => ∫ x, A n x*J (Z n x) ∂μ n) atTop (𝓝 L)) :
∃ η : ProbabilityMeasure (WeightedRestorationRange f C),
(∀ j, (∫ x, x.1.val*x.2.val^j ∂(η : Measure (WeightedRestorationRange f C)))=b j) ∧
(∫ x, x.1.val*J x.2 ∂(η : Measure (WeightedRestorationRange f C)))=L := by
let ν n := compactWeightedLaw (μ n) f C (A n) (hA n) (Z n) (hZ n) (hb n)
obtain ⟨η,-,s,hs,hlim⟩ := isCompact_univ.isSeqCompact (fun n => mem_univ (ν n))
have ht (G : C(WeightedRestorationRange f C,ℝ)) :
Tendsto (fun n => ∫ x, G x ∂(ν (s n) : Measure (WeightedRestorationRange f C))) atTop (𝓝 (∫ x, G x ∂(η : Measure (WeightedRestorationRange f C)))) := by
exact (ProbabilityMeasure.continuous_integral_boundedContinuousFunction
(BoundedContinuousFunction.mkOfCompact G)).continuousAt.tendsto.comp hlim
refine ⟨η,?_,?_⟩
· intro j
let G : C(WeightedRestorationRange f C,ℝ) := ⟨fun x => x.1.val*x.2.val^j,by fun_prop⟩
have hh := ht G
simp only [ν,compactWeightedLaw_integral _ _ _ _ _ _ _ _ _ G.continuous.measurable] at hh
exact tendsto_nhds_unique hh ((hm j).comp hs.tendsto_atTop)
· let G : C(WeightedRestorationRange f C,ℝ) := ⟨fun x => x.1.val*J x.2,
(continuous_subtype_val.comp continuous_fst).mul (J.continuous.comp continuous_snd)⟩
have hh := ht G
simp only [ν,compactWeightedLaw_integral _ _ _ _ _ _ _ _ _ G.continuous.measurable] at hh
exact tendsto_nhds_unique hh (hL.comp hs.tendsto_atTop)
def pairRestorationBound (f v w : ℝ →ᵇ ℝ) (F : CompactOverlap →ᵇ ℝ) : ℝ :=
‖F‖*(∏ _ : Fin 2, ‖expBCF 1 f*v‖)*(∏ _ : Fin 2, ‖expBCF 1 f*w‖)
lemma pairRestorationBound_nonneg (f v w : ℝ →ᵇ ℝ) (F : CompactOverlap →ᵇ ℝ) :
0≤pairRestorationBound f v w F := by unfold pairRestorationBound; positivity
lemma sourceLabelRestorationNumerator_pair_bound (n k : ℕ) (f ψ v w : ℝ →ᵇ ℝ)
(F : CompactOverlap →ᵇ ℝ) (q : Fin (k+1)→Time) (s : ℝ≥0)
(p e : Fin (n+1)→ℕ) (h : Fin (k+1)→ℝ) (u : Fin (n+1)→ℝ) (a) :
|sourceLabelRestorationNumerator n k 2 (fun i => (q i:ℝ)) s f ψ
(fun _ => v) (fun _ => w) (labelProfilePairTest q F) p e h u a|≤pairRestorationBound ψ v w F := by
apply (sourceLabelRestorationNumerator_bound n k 2 _ s f ψ (fun _ => v) (fun _ => w)
(labelProfilePairTest q F) p e h u a).trans
apply mul_le_mul_of_nonneg_right _ (Finset.prod_nonneg fun _ _ => norm_nonneg _)
apply mul_le_mul_of_nonneg_right _ (Finset.prod_nonneg fun _ _ => norm_nonneg _)
exact F.norm_compContinuous_le _
lemma sourceLabelPair_witness (k : ℕ) (q : Fin (k+1)→Time) (hq : Monotone q)
(f : ℝ →ᵇ ℝ) (g : Jet3) (v w : ℝ →ᵇ ℝ) (F : CompactOverlap →ᵇ ℝ)
(p e : (n : ℕ)→Fin (n+1)→ℕ) (h : ℕ→Fin (k+1)→ℝ)
(hh0 : ∀ n i,0≤h n i) (hh : ∀ n,Monotone (h n)) (u : (n : ℕ)→Fin (n+1)→ℝ)
(z : Fin k→ℝ) (hz : StrictMono z) (hz0 : ∀ i,0<z i) (hz1 : ∀ i,z i<1)
(t : ℕ→ℝ≥0) (s : ℕ→ℕ) {ν : ProbabilityMeasure (CompactArray CompactJointOverlap)}
(hlim : Tendsto (fun n => sourceGibbsArrayLaw (s n) k f (p (s n)) (e (s n)) (h (s n))
(u (s n)) z (t (s n))) atTop (𝓝 ν)) :
∃ η : ProbabilityMeasure (WeightedRestorationRange g.f (pairRestorationBound g.f v w F)),
(∀ j, (∫ x, x.1.val*x.2.val^j ∂(η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f v w F))))=
∫ Q, labelRestorationBlockMoment (fun i => (q i:ℝ)) g.f (fun _ => v) (fun _ => w)
(labelProfilePairTest q F) j (compactBlock (2+j) Q) ∂ν) ∧
(∫ x, x.1.val*restorationReciprocal g.f 2 x.2 ∂(η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f v w F))))=
∑ d, F (timeSpin (q d))*(twoVisitMass k z d).toReal*
(labelPairCoefficient k z (fun i => (q i:ℝ)) g (⟨1-q (Fin.last k),sub_nonneg.mpr (q (Fin.last k)).prop.2⟩ : ℝ≥0) v d *
labelPairCoefficient k z (fun i => (q i:ℝ)) g (⟨1-q (Fin.last k),sub_nonneg.mpr (q (Fin.last k)).prop.2⟩ : ℝ≥0) w d) := by
let τ : ℝ≥0 := ⟨1-q (Fin.last k),sub_nonneg.mpr (q (Fin.last k)).prop.2⟩
apply compactWeighted_witness
(fun n => ((sourceBaseDataLaw (s n) k z (t (s n))).prod countableGaussianLaw).prod (countableGaussianLaw.prod countableGaussianLaw))
g.f (pairRestorationBound g.f v w F)
(fun n => sourceLabelRestorationNumerator (s n) k 2 (fun i => (q i:ℝ)) τ f g.f (fun _ => v) (fun _ => w)
(labelProfilePairTest q F) (p (s n)) (e (s n)) (h (s n)) (u (s n)))
(fun n => sourceLabelRestorationDenominator (s n) k (fun i => (q i:ℝ)) τ f g.f
(p (s n)) (e (s n)) (h (s n)) (u (s n)))
(fun n => sourceLabelRestorationNumerator_measurable ..)
(fun n => sourceLabelRestorationDenominator_measurable ..)
(fun n a => sourceLabelRestorationNumerator_pair_bound ..) _ ?_ (restorationReciprocal g.f 2) _ ?_
· intro j
simp_rw [sourceLabelRestorationPolynomial _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (hh0 _) (hh _) j]
exact sourceLabelRestorationMoment_tendsto k 2 (fun i => (q i:ℝ)) (q 0).prop.1
(fun i l hil => hq hil) (q (Fin.last k)).prop.2 f g.f (fun _ => v) (fun _ => w)
(labelProfilePairTest q F) p e h hh0 hh u z t s hlim j
· change Tendsto (fun n => sourceLabelRestorationRatio (s n) k 2 (fun i => (q i:ℝ)) τ f g.f
(fun _ => v) (fun _ => w) (labelProfilePairTest q F) (p (s n)) (e (s n)) (h (s n)) (u (s n)) z (t (s n))) atTop _
simp_rw [sourceLabelRestorationRatio_pair_value _ k f g _ _ _ (hh0 _) (hh _) _ q τ v w F z hz hz0 hz1]
exact tendsto_const_nhds
lemma labelPairCoefficient_one (k : ℕ) (z : Fin k→ℝ) (hz0 : ∀ i,0<z i)
(q : Fin (k+1)→ℝ) (g : Jet3) (s : ℝ≥0) (d : Fin (k+1)) :
labelPairCoefficient k z q g s 1 d=1 := by
let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)) : Measure (EuclideanSpace ℝ (Fin 1))) :=
(inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ (Fin 1))))
have hI := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)))
(labelGaussianStep q) (labelProfileTerminal_lipschitz g s) k z hz0
have hκ i := cascadeTiltedStep_markov (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)))
(gaussianLinearMarkStep (labelGaussianStep q)) (gaussianLinearMarkStep_measurable _)
k z hz0 (fun x => labelProfileTerminal g s (x 0))
((labelProfileTerminal_lipschitz g s).continuous.measurable.comp (measurable_pi_apply 0)) hI.2 i
simp only [labelPairCoefficient,labelResidualObservable,residualResponse_one]
exact rootPathCorrelation_one _ _ k _ hκ d
def labelSingleCoefficient (k : ℕ) (z : Fin k→ℝ) (q : Fin (k+1)→ℝ)
(g : Jet3) (s : ℝ≥0) (v : ℝ →ᵇ ℝ) : ℝ :=
rootPathMean (stdGaussian (EuclideanSpace ℝ (Fin 1)))
(fun r => fun _ => labelGaussianRoot q r) k
(fun i => tiltedStateStep (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)))
(gaussianLinearMarkStep (labelGaussianStep q)) (z i)
(finiteCascadeShifts (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)))
(gaussianLinearMarkStep (labelGaussianStep q)) k z (fun x => labelProfileTerminal g s (x 0)) i))
(fun x => labelResidualObservable s g.f v (x 0))
lemma labelPairCoefficient_bounds (k : ℕ) (z : Fin k→ℝ) (hz0 : ∀ i,0<z i)
(q : Fin (k+1)→ℝ) (g : Jet3) (s : ℝ≥0) (v : ℝ →ᵇ ℝ) :
(∀ d,0≤labelPairCoefficient k z q g s v d) ∧
Monotone (labelPairCoefficient k z q g s v) ∧
(∀ d, labelPairCoefficient k z q g s v d≤labelSingleCoefficient k z q g s (v*v)) ∧
labelSingleCoefficient k z q g s (v*v)≤‖v‖^2 := by
let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)) : Measure (EuclideanSpace ℝ (Fin 1))) :=
(inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ (Fin 1))))
have hI := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)))
(labelGaussianStep q) (labelProfileTerminal_lipschitz g s) k z hz0
have hκ i := cascadeTiltedStep_markov (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin 1)))
(gaussianLinearMarkStep (labelGaussianStep q)) (gaussianLinearMarkStep_measurable _)
k z hz0 (fun x => labelProfileTerminal g s (x 0))
((labelProfileTerminal_lipschitz g s).continuous.measurable.comp (measurable_pi_apply 0)) hI.2 i
exact residual_path_pair_coefficients (stdGaussian (EuclideanSpace ℝ (Fin 1)))
(fun r => fun _ => labelGaussianRoot q r) (by fun_prop) k _ hκ
(fun x => labelEvaluation (x 0)) (by unfold labelEvaluation; fun_prop) s g.f v
end SphericalPerceptronFreeEnergy
end
end OAI
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