openAI-math-dataset / lean /OAI /Probability /Perceptron /Variational /CompactRestorationWitness.lean
| 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 | |