openAI-math-dataset / lean /OAI /Probability /Perceptron /Variational /CountableSingleRestoration.lean
| import Mathlib | |
| import OAI.Probability.Perceptron.Interpolation.TripleGaussianReplica | |
| import OAI.Probability.Perceptron.Variational.SingleRestorationLaw | |
| namespace OAI | |
| noncomputable section | |
| namespace SphericalPerceptronFreeEnergy | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal NNReal BigOperators | |
| variable {I J K : Type} [Fintype I] [Fintype J] [Fintype K] | |
| def tripleGaussianMean (k : ℕ) | |
| (A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ) | |
| (f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ) | |
| (p : TripleGaussianData I J K k) : ℝ := | |
| indexedTerminalMean (tripleGaussianStep A B C) k (tripleGaussianTerminal H G F) | |
| (tripleGaussianRoot R Q T p.1) (tripleGaussianObservable f g) | |
| (p.2.1,indexedMarksZip k p.2.2) | |
| lemma tripleGaussianMean_measurable (k : ℕ) | |
| (A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ) | |
| (hH : Measurable H) (hG : Measurable G) (hF : Measurable F) | |
| (f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ) | |
| (hf : Measurable f) (hg : Measurable g) : | |
| Measurable (tripleGaussianMean k A B C R Q T H G F f g) := by | |
| have ht : Measurable (tripleGaussianTerminal H G F) := by unfold tripleGaussianTerminal; fun_prop | |
| have ho : Measurable (tripleGaussianObservable (K:=K) f g) := by unfold tripleGaussianObservable; fun_prop | |
| have hr : Measurable (tripleGaussianRoot R Q T) := by unfold tripleGaussianRoot; fun_prop | |
| have hm := indexedTerminalMean_joint_measurable (tripleGaussianStep A B C) | |
| (tripleGaussianStep_measurable A B C) k (tripleGaussianTerminal H G F) ht | |
| (tripleGaussianObservable f g) ho | |
| have hp : Measurable (fun p : TripleGaussianData I J K k => | |
| (tripleGaussianRoot R Q T p.1,(p.2.1,indexedMarksZip k p.2.2))) := | |
| (hr.comp measurable_fst).prodMk (measurable_snd.fst.prodMk | |
| ((indexedMarksZip_measurable k).comp measurable_snd.snd)) | |
| have hc := hm.comp hp | |
| exact hc | |
| theorem countable_three_terminal_single_value (k : ℕ) (z : Fin k→ℝ) | |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) | |
| (A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ) | |
| {L₁ L₂ L₀ : ℝ≥0} (hH : LipschitzWith L₁ H) (hG : LipschitzWith L₂ G) (hF : LipschitzWith L₀ F) | |
| (f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ) | |
| (hf : Measurable f) (hg : Measurable g) (D E : ℝ) (hD : 0≤D) (hE : 0≤E) | |
| (hfB : ∀ x, |f x|≤D) (hgB : ∀ y, |g y|≤E) : | |
| (∫ t : IndexedCascadeBase k×(((ℕ→ℝ)×(ℕ→ℝ))×(ℕ→ℝ)), | |
| tripleGaussianMean k A B C R Q T H G F f g | |
| ((((indexedGaussianDisorder k I t.2.1.1).1,(indexedGaussianDisorder k J t.2.1.2).1), | |
| (indexedGaussianDisorder k K t.2.2).1), | |
| (t.1,(indexedMarksZip k ((indexedGaussianDisorder k I t.2.1.1).2, | |
| (indexedGaussianDisorder k J t.2.1.2).2),(indexedGaussianDisorder k K t.2.2).2))) | |
| ∂(indexedCascadeBaseLaw k z : Measure (IndexedCascadeBase k)).prod | |
| ((countableGaussianLaw.prod countableGaussianLaw).prod countableGaussianLaw)) = | |
| (rootPathMean (stdGaussian (EuclideanSpace ℝ I)) (fun r => fun _ => R r) k | |
| (fun i => tiltedStateStep (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianLinearMarkStep A) (z i) | |
| (finiteCascadeShifts (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianLinearMarkStep A) k z (fun x => H (x 0)) i)) | |
| (fun x => f (x 0)) * | |
| rootPathMean (stdGaussian (EuclideanSpace ℝ J)) (fun r => fun _ => Q r) k | |
| (fun i => tiltedStateStep (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) (gaussianLinearMarkStep B) (z i) | |
| (finiteCascadeShifts (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) (gaussianLinearMarkStep B) k z (fun x => G (x 0)) i)) | |
| (fun x => g (x 0))) := by | |
| let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ I) : Measure (EuclideanSpace ℝ I)) := (inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ I))) | |
| let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ J) : Measure (EuclideanSpace ℝ J)) := (inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ J))) | |
| let : IsGaussian (gaussianMarkLaw (E:=EuclideanSpace ℝ K) : Measure (EuclideanSpace ℝ K)) := (inferInstance : IsGaussian (stdGaussian (EuclideanSpace ℝ K))) | |
| have hI := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) A hH k z hz0 | |
| have hJ := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) B hG k z hz0 | |
| have hK := gaussianLinearRecursion_realization (gaussianMarkLaw (E:=EuclideanSpace ℝ K)) C hF k z hz0 | |
| have hv := indexedTerminalMean_two_fresh_old | |
| (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) | |
| (gaussianMarkLaw (E:=EuclideanSpace ℝ K)) | |
| (gaussianMarkLaw (E:=EuclideanSpace ℝ I)) (gaussianMarkLaw (E:=EuclideanSpace ℝ J)) | |
| (gaussianMarkLaw (E:=EuclideanSpace ℝ K)) | |
| (gaussianLinearMarkStep A) (gaussianLinearMarkStep B) (gaussianLinearMarkStep C) | |
| (gaussianLinearMarkStep_measurable A) (gaussianLinearMarkStep_measurable B) (gaussianLinearMarkStep_measurable C) | |
| k z hz hz0 hz1 (fun x => H (x 0)) (fun x => G (x 0)) (fun x => F (x 0)) | |
| (hH.continuous.measurable.comp (measurable_pi_apply 0)) | |
| (hG.continuous.measurable.comp (measurable_pi_apply 0)) | |
| (hF.continuous.measurable.comp (measurable_pi_apply 0)) hI.2 hJ.2 hK.2 | |
| (fun r => fun _ => R r) (fun r => fun _ => Q r) (fun r => fun _ => T r) | |
| (by fun_prop) (by fun_prop) (by fun_prop) (fun x => f (x 0)) (fun x => g (x 0)) | |
| (hf.comp (measurable_pi_apply 0)) (hg.comp (measurable_pi_apply 0)) D E hD hE (fun x => hfB _) (fun x => hgB _) | |
| have hp := indexedGaussian_three_regroup_law k z I J K | |
| have hm := tripleGaussianMean_measurable k A B C R Q T H G F hH.continuous.measurable | |
| hG.continuous.measurable hF.continuous.measurable f g hf hg | |
| have he := integral_map (μ:=(indexedCascadeBaseLaw k z : Measure (IndexedCascadeBase k)).prod | |
| ((countableGaussianLaw.prod countableGaussianLaw).prod countableGaussianLaw)) | |
| hp.measurable.aemeasurable hm.aestronglyMeasurable | |
| rw [hp.map_eq] at he | |
| exact he.symm.trans hv | |
| lemma tripleGaussianMean_replica (k : ℕ) | |
| (A : ℕ→EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (B : ℕ→EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (C : ℕ→EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (R : EuclideanSpace ℝ I →L[ℝ] EuclideanSpace ℝ I) | |
| (Q : EuclideanSpace ℝ J →L[ℝ] EuclideanSpace ℝ J) | |
| (T : EuclideanSpace ℝ K →L[ℝ] EuclideanSpace ℝ K) | |
| (H : EuclideanSpace ℝ I→ℝ) (G : EuclideanSpace ℝ J→ℝ) (F : EuclideanSpace ℝ K→ℝ) | |
| (hH : Measurable H) (hG : Measurable G) (hF : Measurable F) | |
| (f : EuclideanSpace ℝ I→ℝ) (g : EuclideanSpace ℝ J→ℝ) | |
| (b : IndexedCascadeBase k) (t : ((ℕ→ℝ)×(ℕ→ℝ))×(ℕ→ℝ)) | |
| (hi : Integrable (fun l => Real.exp ((H (gaussianLeafValue k A R t.1.1 l)+ | |
| G (gaussianLeafValue k B Q t.1.2 l))+F (gaussianLeafValue k C T t.2 l))) (indexedLeafProbability k b)) : | |
| tripleGaussianMean k A B C R Q T H G F f g (tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t) = | |
| gibbsReplicaMean (indexedLeafProbability k b) | |
| (fun l => (H (gaussianLeafValue k A R t.1.1 l)+G (gaussianLeafValue k B Q t.1.2 l))+ | |
| F (gaussianLeafValue k C T t.2 l)) 1 | |
| (fun l => f (gaussianLeafValue k A R t.1.1 (l 0))*g (gaussianLeafValue k B Q t.1.2 (l 0))) := by | |
| have ht : Measurable (tripleGaussianTerminal H G F) := by unfold tripleGaussianTerminal; fun_prop | |
| unfold tripleGaussianMean | |
| have he := indexedTerminalMean_tiltMean (tripleGaussianStep A B C) (tripleGaussianStep_measurable A B C) | |
| k (tripleGaussianTerminal H G F) ht | |
| (tripleGaussianRoot R Q T (tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t).1) (tripleGaussianObservable f g) | |
| ((tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t).2.1,indexedMarksZip k (tripleGaussianCountableData (I:=I) (J:=J) (K:=K) k b t).2.2) | |
| have hstate (l : IndexedLeaf k) := tripleGaussianLeafState k A B C | |
| (fun _ => R (indexedGaussianDisorder k I t.1.1).1) | |
| (fun _ => Q (indexedGaussianDisorder k J t.1.2).1) | |
| (fun _ => T (indexedGaussianDisorder k K t.2).1) | |
| (indexedGaussianDisorder k I t.1.1).2 (indexedGaussianDisorder k J t.1.2).2 | |
| (indexedGaussianDisorder k K t.2).2 l | |
| simp only [tripleGaussianCountableData,tripleGaussianRoot] at he ⊢ | |
| simp only [tripleGaussianTerminal,hstate] at he | |
| have hv := he hi | |
| simp only [tripleGaussianObservable] at hv | |
| rw [hv] | |
| symm | |
| refine (gibbsReplicaMean_coordinate_of_integrable (indexedLeafProbability k b) | |
| (F:=fun l => f (gaussianLeafValue k A R t.1.1 l)*g (gaussianLeafValue k B Q t.1.2 l)) | |
| (measurable_of_countable _) (measurable_of_countable _) hi 1 0).trans ?_ | |
| congr 1 | |
| funext l | |
| simp only [indexedTerminalEnergy,tripleGaussianTerminal,hstate,gaussianLeafValue] | |
| end SphericalPerceptronFreeEnergy | |
| end | |
| end OAI | |