| import Mathlib |
| import OAI.Probability.Perceptron.Control.FreshControl |
| import OAI.Probability.Perceptron.Cavity.BulkCompactGG |
| import OAI.Probability.Perceptron.Cavity.BulkRestorationLimit |
| import OAI.Probability.Perceptron.Variational.LabelProfilePairReference |
| import OAI.Probability.Perceptron.Interpolation.GibbsArrayGeometry |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology BigOperators BoundedContinuousFunction NNReal ENNReal |
|
|
| def bulkJointEmbedding (q : CompactOverlap) : CompactJointOverlap := (q,0) |
| lemma bulkJointEmbedding_continuous : Continuous bulkJointEmbedding := by |
| unfold bulkJointEmbedding; fun_prop |
|
|
| def bulkJointLaw (μ : ProbabilityMeasure (CompactArray CompactOverlap)) : |
| ProbabilityMeasure (CompactArray CompactJointOverlap) := |
| compactMapLaw bulkJointEmbedding bulkJointEmbedding_continuous μ |
|
|
| lemma bulkGibbsArray_exchangeable (n M : ℕ) (f : ℝ→ᵇℝ) (v : ℕ→ℝ) (e : Equiv.Perm ℕ) : |
| MeasurePreserving (compactRelabel (K:=CompactOverlap) e) |
| (bulkGibbsArrayLaw n M f v : Measure _) (bulkGibbsArrayLaw n M f v : Measure _) := |
| gibbsOverlapArray_exchangeable _ _ _ _ _ _ e |
|
|
| lemma bulkGibbsArray_closed_full (n M : ℕ) (f : ℝ→ᵇℝ) (v : ℕ→ℝ) |
| {F : Set (CompactArray CompactOverlap)} (hF : IsClosed F) |
| (hfull : ∀ x : ℕ→NormalizedSpin (n+1), overlapArray bulkOverlap x ∈ F) : |
| bulkGibbsArrayLaw n M f v F=1 := |
| gibbsOverlapArray_closed_full _ _ _ _ _ _ hF hfull |
|
|
| lemma compact_scalar_gram_closed (r : ℕ) : |
| IsClosed {Q : CompactArray CompactOverlap | Matrix.PosSemidef (fun i j : Fin r => (Q i j).val)} := |
| (isClosed_matrix_posSemidef r).preimage (by fun_prop) |
|
|
| lemma bulkGibbsArray_limit_gram (f : ℝ→ᵇℝ) (M : ℕ→ℕ) (v : ℕ→ℕ→ℝ) (s : ℕ→ℕ) |
| {ν : ProbabilityMeasure (CompactArray CompactOverlap)} |
| (hlim : Tendsto (fun n => bulkGibbsArrayLaw (s n) (M (s n)) f (v (s n))) atTop (𝓝 ν)) : |
| ∀ᵐ Q : CompactArray CompactOverlap ∂(ν : Measure _), ∀ r, |
| Matrix.PosSemidef (fun i j : Fin r => (Q i j).val) := by |
| rw [ae_all_iff] |
| intro r |
| apply (mem_ae_iff_prob_eq_one (compact_scalar_gram_closed r).measurableSet).mpr |
| have hv := weak_limit_closed_full hlim (compact_scalar_gram_closed r) |
| (fun n => bulkGibbsArray_closed_full (s n) (M (s n)) f (v (s n)) |
| (compact_scalar_gram_closed r) (fun x => Matrix.posSemidef_gram ℝ (fun i : Fin r => (x i).val))) |
| simp only [← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure,hv,ENNReal.coe_one] |
|
|
| lemma bulkGibbsArray_limit_diagonal (f : ℝ→ᵇℝ) (M : ℕ→ℕ) (v : ℕ→ℕ→ℝ) (s : ℕ→ℕ) |
| {ν : ProbabilityMeasure (CompactArray CompactOverlap)} |
| (hlim : Tendsto (fun n => bulkGibbsArrayLaw (s n) (M (s n)) f (v (s n))) atTop (𝓝 ν)) : |
| ∀ᵐ Q : CompactArray CompactOverlap ∂(ν : Measure _), ∀ i, (Q i i).val=1 := by |
| rw [ae_all_iff] |
| intro i |
| have hc : IsClosed {Q : CompactArray CompactOverlap | (Q i i).val=1} := |
| isClosed_eq (by fun_prop) continuous_const |
| apply (mem_ae_iff_prob_eq_one hc.measurableSet).mpr |
| have hv := weak_limit_closed_full hlim hc |
| (fun n => bulkGibbsArray_closed_full (s n) (M (s n)) f (v (s n)) hc |
| (fun x => spinOverlap_self (x i))) |
| simp only [← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure,hv,ENNReal.coe_one] |
|
|
| lemma bulkJointLaw_gram {ν : ProbabilityMeasure (CompactArray CompactOverlap)} |
| (hR : ∀ᵐ Q : CompactArray CompactOverlap ∂(ν : Measure _), ∀ r, |
| Matrix.PosSemidef (fun i j : Fin r => (Q i j).val)) : |
| ∀ᵐ Q : CompactArray CompactJointOverlap ∂(bulkJointLaw ν : Measure _), ∀ r, |
| Matrix.PosSemidef (fun i j : Fin r => (Q i j).1.val) := by |
| apply (ae_map_iff (compactMapArray_continuous bulkJointEmbedding_continuous).measurable.aemeasurable |
| (show MeasurableSet {Q : CompactArray CompactJointOverlap | ∀ r, |
| Matrix.PosSemidef (fun i j : Fin r => (Q i j).1.val)} by |
| simp only [ofPred_forall] |
| exact MeasurableSet.iInter fun r => (compact_spin_gram_closed r).measurableSet)).mpr |
| exact hR |
|
|
| lemma bulkJointLaw_diagonal {ν : ProbabilityMeasure (CompactArray CompactOverlap)} |
| (hd : ∀ᵐ Q : CompactArray CompactOverlap ∂(ν : Measure _), ∀ i, (Q i i).val=1) : |
| ∀ᵐ Q : CompactArray CompactJointOverlap ∂(bulkJointLaw ν : Measure _), ∀ i, (Q i i).1.val=1 := by |
| apply (ae_map_iff (compactMapArray_continuous bulkJointEmbedding_continuous).measurable.aemeasurable |
| (show MeasurableSet {Q : CompactArray CompactJointOverlap | ∀ i, (Q i i).1.val=1} by |
| simp only [ofPred_forall] |
| exact MeasurableSet.iInter fun i => measurableSet_eq_fun (by fun_prop) measurable_const)).mpr |
| exact hd |
|
|
| lemma omitted_bulk_limit_geometry (α : ℝ) (f : ℝ→ᵇℝ) (v : ℕ→ℕ→ℝ) |
| (hv : ∀ᶠ n in atTop, ∀ p, 1≤v n p) |
| (hd : ∀ p, Tendsto (fun n => omittedBulkDeviation α f n p (v n)) atTop (𝓝 0)) |
| {ν : ProbabilityMeasure (CompactArray CompactOverlap)} {s : ℕ→ℕ} (hs : StrictMono s) |
| (hlim : Tendsto (fun n => bulkGibbsArrayLaw (s n) (patternCount α (s n+1)-2) f (v (s n))) atTop (𝓝 ν)) : |
| (∀ᵐ Q ∂(bulkJointLaw ν : Measure (CompactArray CompactJointOverlap)), CompactSpinGeometry Q) ∧ |
| (∀ᵐ Q : CompactArray CompactJointOverlap ∂(bulkJointLaw ν : Measure _), 0≤(Q 0 1).1.val) ∧ |
| (∀ r (i : Fin r) (F : CompactBlock CompactJointOverlap r→ᵇℝ) (g : CompactJointOverlap→ᵇℝ), |
| compactGGDefect (bulkJointLaw ν) r i F g=0) := by |
| have hGG := compactMapLaw_gg bulkJointEmbedding bulkJointEmbedding_continuous ν |
| (omitted_bulk_limit_gg α f v hv hd hs hlim) |
| have hR := bulkJointLaw_gram (bulkGibbsArray_limit_gram f (fun n => patternCount α (n+1)-2) v s hlim) |
| have hD := bulkJointLaw_diagonal (bulkGibbsArray_limit_diagonal f (fun n => patternCount α (n+1)-2) v s hlim) |
| have hEx (e : Equiv.Perm ℕ) := compactMapLaw_exchangeable bulkJointEmbedding bulkJointEmbedding_continuous ν e |
| (compact_exchangeability_limit hlim e (fun n => bulkGibbsArray_exchangeable (s n) |
| (patternCount α (s n+1)-2) f (v (s n)) e)) |
| exact ⟨compact_gg_spin_geometry _ hEx hGG hR hD, |
| compact_spin_nonnegative _ (compactRealLaw_gg _ hGG) hR,hGG⟩ |
|
|
| lemma bulkJointLaw_scalarGram_integral (ν : ProbabilityMeasure (CompactArray CompactOverlap)) |
| (r : ℕ) (F : Matrix (Fin r) (Fin r) ℝ →ᵇ ℝ) : |
| (∫ Q : CompactArray CompactJointOverlap, F (fun i j => (Q i j).1.val) ∂(bulkJointLaw ν : Measure _))= |
| ∫ Q : CompactArray CompactOverlap, F (fun i j => (Q i j).val) ∂(ν : Measure _) := by |
| exact compactMapLaw_integral bulkJointEmbedding bulkJointEmbedding_continuous ν |
| (fun Q => F (fun i j => (Q i j).1.val)) (F.continuous.comp (by fun_prop)).measurable |
|
|
| lemma bulk_pair_restoration_ratio_tendsto (M : ℕ→ℕ) (g : Jet3) (v : ℕ→ℕ→ℝ) (s : ℕ→ℕ) |
| {ν : ProbabilityMeasure (CompactArray CompactOverlap)} |
| (hlim : Tendsto (fun n => bulkGibbsArrayLaw (s n) (M (s n)) g.f (v (s n))) atTop (𝓝 ν)) |
| (hGG : ∀ (r : ℕ) (i : Fin r) (G : CompactBlock CompactJointOverlap r →ᵇ ℝ) |
| (a : CompactJointOverlap →ᵇ ℝ), compactGGDefect (bulkJointLaw ν) r i G a=0) |
| (hgeo : ∀ᵐ Q ∂(bulkJointLaw ν : Measure (CompactArray CompactJointOverlap)), CompactSpinGeometry Q) |
| (hn : ∀ᵐ Q ∂(bulkJointLaw ν : Measure (CompactArray CompactJointOverlap)), 0≤(Q 0 1).1.val) |
| (F : CompactOverlap →ᵇ ℝ) : |
| Tendsto (fun n => ∫ p, bulkFreshNumerator (s n) (M (s n)) 2 g.f (v (s n)) |
| (compactPairTest F) (gaussianMixedTest (fun _ => 1)) (gaussianMixedTest (fun _ => 1)) p / |
| bulkFreshDenominator (s n) (M (s n)) g.f (v (s n)) p^2 |
| ∂twoFreshDisorderLaw (s n+1) (M (s n))) atTop |
| (𝓝 (∫ Q : CompactArray CompactOverlap, F (Q 0 1) ∂(ν : Measure _))) := by |
| let q := boundedQuantile ((compactPositivePairLaw (bulkJointLaw ν)).map Prod.fst) |
| obtain ⟨ρ,η,hG,hge,hp,hm,hl⟩ := quantile_pair_reference_exists q (boundedQuantile_monotone _) g F |
| have hpair := (compactPositivePairLaw_spin_quantile (bulkJointLaw ν) hn).trans hp.symm |
| have he (j : ℕ) : (∫ x, x.1.val*x.2.val^j |
| ∂(η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f 1 1 F))))= |
| ∫ Q : CompactArray CompactOverlap, scalarRestorationPairMoment g.f 1 1 F j |
| (fun i l => (Q i l).val) ∂(ν : Measure _) := by |
| rw [hm j,←scalarGram_eq_of_pair_law (bulkJointLaw ν) ρ hGG hG hgeo hge hpair] |
| exact bulkJointLaw_scalarGram_integral ν _ _ |
| let Ψ : EuclideanSpace ℝ (Fin 2) →ᵇ ℝ := gaussianMixedTest (fun _ => (1 : ℝ →ᵇ ℝ)) |
| let C := ‖compactPairTest F‖*‖restorationMarkTest 2 g.f Ψ 0‖*‖restorationMarkTest 2 g.f Ψ 0‖+ |
| pairRestorationBound g.f 1 1 F |
| have hC : 0≤C := add_nonneg (by positivity) (pairRestorationBound_nonneg _ _ _ _) |
| have hηB (x : WeightedRestorationRange g.f (pairRestorationBound g.f 1 1 F)) : |x.1.val|≤C := |
| (abs_le.mpr x.1.prop).trans (le_add_of_nonneg_left (by positivity)) |
| have hb (n : ℕ) (p : TwoFreshDisorder (s n+1) (M (s n))) : |
| |bulkFreshNumerator (s n) (M (s n)) 2 g.f (v (s n)) (compactPairTest F) Ψ Ψ p|≤C := |
| (bulkFreshNumerator_bound _ _ _ _ _ _ _ _ _).trans |
| (le_add_of_nonneg_right (pairRestorationBound_nonneg _ _ _ _)) |
| have hmoment j : Tendsto (fun n => ∫ p, |
| bulkFreshNumerator (s n) (M (s n)) 2 g.f (v (s n)) (compactPairTest F) Ψ Ψ p * |
| (bulkFreshCompactDenominator (s n) (M (s n)) g.f (v (s n)) p).val^j |
| ∂twoFreshDisorderLaw (s n+1) (M (s n))) atTop |
| (𝓝 (∫ x,x.1.val*x.2.val^j ∂(η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f 1 1 F))))) := by |
| rw [he j] |
| simp_rw [scalarRestorationPairMoment_compact] |
| exact bulkFresh_moment_tendsto M g.f v s hlim 2 (compactPairTest F) Ψ Ψ j |
| have ht := restoration_ratio_tendsto |
| (fun n => twoFreshDisorderLaw (s n+1) (M (s n))) |
| (η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f 1 1 F))) |
| (Real.exp_pos _) |
| (fun n => bulkFreshNumerator (s n) (M (s n)) 2 g.f (v (s n)) (compactPairTest F) Ψ Ψ) |
| (fun x => x.1.val) |
| (fun n => bulkFreshCompactDenominator (s n) (M (s n)) g.f (v (s n))) (fun x => x.2) |
| (fun n => bulkFreshNumerator_measurable _ _ _ _ _ _ _ _) (by fun_prop) |
| (fun n => bulkFreshCompactDenominator_measurable _ _ _ _) (by fun_prop) |
| hC hb hηB hmoment 2 |
| have hvalue : (∫ x,x.1.val/x.2.val^2 |
| ∂(η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f 1 1 F))))= |
| ∫ Q : CompactArray CompactOverlap,F (Q 0 1) ∂(ν : Measure _) := by |
| simp only [div_eq_mul_inv] |
| change (∫ x,x.1.val*restorationReciprocal g.f 2 x.2 ∂(η : Measure (WeightedRestorationRange g.f (pairRestorationBound g.f 1 1 F))))=_ |
| rw [hl,←compactPositivePairLaw_spin_quantile (bulkJointLaw ν) hn] |
| have hi := integral_map (μ := (bulkJointLaw ν : Measure (CompactArray CompactJointOverlap))) |
| (show Measurable (fun Q : CompactArray CompactJointOverlap => (Q 0 1).1) |
| from by fun_prop).aemeasurable |
| (show Measurable (fun z : CompactOverlap => F z) from F.continuous.measurable).aestronglyMeasurable |
| refine hi.trans ?_ |
| exact compactMapLaw_integral bulkJointEmbedding bulkJointEmbedding_continuous ν |
| (fun Q => F (Q 0 1).1) (by fun_prop) |
| rw [hvalue] at ht |
| exact ht |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
|
|
