| import Mathlib |
| import OAI.Probability.Perceptron.Cavity.BulkGoodParameters |
| import OAI.Probability.Perceptron.Variational.CompactScalarGG |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma bulkOverlap_continuous (N : ℕ) : Continuous (Function.uncurry (bulkOverlap (N:=N))) := by |
| apply Continuous.subtype_mk |
| exact continuous_fst.subtype_val.inner continuous_snd.subtype_val |
|
|
| lemma bulkBlockTest_measurable (N r : ℕ) (F : CompactBlock CompactOverlap r →ᵇ ℝ) : |
| Measurable (fun x : Fin r→NormalizedSpin N => F (fun i j => bulkOverlap (x i) (x j))) := by |
| apply F.continuous.measurable.comp |
| refine Measurable.of_eval fun first => Measurable.of_eval fun second => ?_ |
| have hm : Measurable (fun x : Fin r → NormalizedSpin N => (x first, x second)) := |
| (measurable_pi_apply first).prodMk (measurable_pi_apply second) |
| exact (bulkOverlap_measurable N).comp hm |
|
|
| lemma bulk_compact_gg_defect (n M : ℕ) (f : ℝ→ᵇℝ) (v : ℕ→ℝ) (p r : ℕ) (j : Fin r) |
| (F : CompactBlock CompactOverlap r →ᵇ ℝ) : |
| compactGGDefect (bulkGibbsArrayLaw n M f v) r j F (compactScalarMonomial p) = |
| -bulkGGDefect n M f v p r j (fun x => F (fun i l => bulkOverlap (x i) (x l))) := by |
| have hh := compactGGDefect_gibbsOverlapArray (bulkSpinKernel n M) (bulkDisorderLaw (n+1) M) |
| (fun a x => bulkHamiltonian (n+1) M f v a.1 a.2 x) |
| (bulkHamiltonian_continuous _ _ _ _).measurable bulkOverlap (bulkOverlap_measurable (n+1)) |
| (ae_of_all _ (bulkHamiltonian_exp_integrable n M f v)) r j F (compactScalarMonomial p) 1 |
| (fun x => by simp only [compactScalarMonomial_apply,bulkOverlap,spinOverlap_self,one_pow]) |
| (fun x y => by simp only [compactScalarMonomial_apply,bulkOverlap,spinOverlap,real_inner_comm]) |
| simpa only [bulkGibbsArrayLaw,kernelGGDefect,bulkGGDefect,bulkMoment,bulkSpinKernel, |
| Kernel.const_apply,compactScalarMonomial_apply,bulkOverlap,one_mul] using hh |
|
|
| lemma bulk_compact_gg_bound (n M : ℕ) (f : ℝ→ᵇℝ) (v : ℕ→ℝ) (p : Fin (n+1)) (r : ℕ) (j : Fin r) |
| (F : CompactBlock CompactOverlap r →ᵇ ℝ) (hv : 1≤v (p.val+1)) : |
| |compactGGDefect (bulkGibbsArrayLaw n M f v) r j F (compactScalarMonomial (p.val+1))| ≤ |
| ‖F‖/(2:ℝ)^(-((p.val+1:ℕ):ℤ))*bulkNormalizedDeviation n M f p v := by |
| rw [bulk_compact_gg_defect,abs_neg] |
| exact bulkGG_bound n M f v p r j (bulkBlockTest_measurable (n+1) r F) (norm_nonneg _) |
| (fun x => by simpa only [Real.norm_eq_abs] using F.norm_coe_le_norm (fun i l => bulkOverlap (x i) (x l))) hv |
|
|
| lemma omitted_bulk_monomial_tendsto (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)) |
| (p r : ℕ) (j : Fin r) (F : CompactBlock CompactOverlap r →ᵇ ℝ) : |
| Tendsto (fun n => compactGGDefect (bulkGibbsArrayLaw n (patternCount α (n+1)-2) f (v n)) |
| r j F (compactScalarMonomial (p+1))) atTop (𝓝 0) := by |
| apply squeeze_zero_norm' (a:=fun n => ‖F‖/(2:ℝ)^(-((p+1:ℕ):ℤ))*omittedBulkDeviation α f n p (v n)) |
| · filter_upwards [hv,eventually_ge_atTop p] with n hn hp |
| have hpn : p<n+1 := Nat.lt_succ_of_le hp |
| simpa only [Real.norm_eq_abs,omittedBulkDeviation,bulkDeviationAt,dite_eq_left hpn] using |
| bulk_compact_gg_bound n (patternCount α (n+1)-2) f (v n) ⟨p,hpn⟩ r j F (hn (p+1)) |
| · simpa using (hd p).const_mul (‖F‖/(2:ℝ)^(-((p+1:ℕ):ℤ))) |
| |
| lemma compactScalarGG_zero (μ : ProbabilityMeasure (CompactArray CompactOverlap)) |
| (r : ℕ) (j : Fin r) (F : CompactBlock CompactOverlap r →ᵇ ℝ) : |
| compactGGDefect μ r j F (compactScalarMonomial 0)=0 := by |
| have hr : 1≤r := by have := j.isLt; omega |
| simp only [compactGGDefect,compactScalarMonomial_apply,pow_zero,mul_one, |
| integral_const,probReal_univ,smul_eq_mul,Finset.sum_const,Finset.card_erase_of_mem |
| (Finset.mem_univ j),Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_sub hr,Nat.cast_one] |
| ring |
| |
| lemma omitted_bulk_limit_gg (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)) |
| (r : ℕ) (j : Fin r) (F : CompactBlock CompactOverlap r →ᵇ ℝ) (g : CompactOverlap →ᵇ ℝ) : |
| compactGGDefect ν r j F g=0 := by |
| apply compactGG_of_scalar_monomials |
| intro p |
| cases p with |
| | zero => exact compactScalarGG_zero ν r j F |
| | succ p => |
| apply compactGGDefect_limit hlim |
| exact (omitted_bulk_monomial_tendsto α f v hv hd p r j F).comp hs.tendsto_atTop |
| |
| |
| lemma bulk_monomial_tendsto (M : ℕ→ℕ) (f : ℝ→ᵇℝ) (v : ℕ→ℕ→ℝ) |
| (hv : ∀ᶠ n in atTop, ∀ p, 1≤v n p) |
| (hd : ∀ p, Tendsto (fun n => bulkDeviationAt n (M n) f p (v n)) atTop (𝓝 0)) |
| (p r : ℕ) (j : Fin r) (F : CompactBlock CompactOverlap r →ᵇ ℝ) : |
| Tendsto (fun n => compactGGDefect (bulkGibbsArrayLaw n (M n) f (v n)) |
| r j F (compactScalarMonomial (p+1))) atTop (𝓝 0) := by |
| apply squeeze_zero_norm' (a:=fun n => ‖F‖/(2:ℝ)^(-((p+1:ℕ):ℤ))*bulkDeviationAt n (M n) f p (v n)) |
| · filter_upwards [hv,eventually_ge_atTop p] with n hn hp |
| have hpn : p<n+1 := Nat.lt_succ_of_le hp |
| simpa only [Real.norm_eq_abs,bulkDeviationAt,dite_eq_left hpn] using |
| bulk_compact_gg_bound n (M n) f (v n) ⟨p,hpn⟩ r j F (hn (p+1)) |
| · simpa using (hd p).const_mul (‖F‖/(2:ℝ)^(-((p+1:ℕ):ℤ))) |
|
|
| lemma bulk_limit_gg (M : ℕ→ℕ) (f : ℝ→ᵇℝ) (v : ℕ→ℕ→ℝ) |
| (hv : ∀ᶠ n in atTop, ∀ p, 1≤v n p) |
| (hd : ∀ p, Tendsto (fun n => bulkDeviationAt n (M n) f p (v n)) atTop (𝓝 0)) |
| {ν : ProbabilityMeasure (CompactArray CompactOverlap)} {s : ℕ→ℕ} (hs : StrictMono s) |
| (hlim : Tendsto (fun n => bulkGibbsArrayLaw (s n) (M (s n)) f (v (s n))) atTop (𝓝 ν)) |
| (r : ℕ) (j : Fin r) (F : CompactBlock CompactOverlap r →ᵇ ℝ) (g : CompactOverlap →ᵇ ℝ) : |
| compactGGDefect ν r j F g=0 := by |
| apply compactGG_of_scalar_monomials |
| intro p |
| cases p with |
| | zero => exact compactScalarGG_zero ν r j F |
| | succ p => |
| apply compactGGDefect_limit hlim |
| exact (bulk_monomial_tendsto M f v hv hd p r j F).comp hs.tendsto_atTop |
|
|
| def bulkJointDeviation (α : ℝ) (f : ℝ→ᵇℝ) (n p : ℕ) (v : ℕ→ℝ) : ℝ := |
| bulkDeviationAt n (patternCount α (n+1)) f p v + omittedBulkDeviation α f n p v |
|
|
| lemma bulkJointDeviation_nonneg (α : ℝ) (f : ℝ→ᵇℝ) (n p : ℕ) (v : ℕ→ℝ) : |
| 0≤bulkJointDeviation α f n p v := |
| add_nonneg (bulkDeviationAt_nonneg _ _ _ _ _) (bulkDeviationAt_nonneg _ _ _ _ _) |
| lemma bulkJointDeviation_measurable (α : ℝ) (f : ℝ→ᵇℝ) (n p : ℕ) : |
| Measurable (bulkJointDeviation α f n p) := |
| (bulkDeviationAt_measurable _ _ _ _).add (bulkDeviationAt_measurable _ _ _ _) |
| lemma bulkJointDeviation_integrable (α : ℝ) (f : ℝ→ᵇℝ) (n p : ℕ) : |
| Integrable (bulkJointDeviation α f n p) bulkParameterLaw := |
| (bulkDeviationAt_integrable _ _ _ _).add (bulkDeviationAt_integrable _ _ _ _) |
|
|
| lemma bulkJointDeviation_integral_tendsto {α : ℝ} (hα : 0≤α) (f : ℝ→ᵇℝ) (p : ℕ) : |
| Tendsto (fun n => ∫ v, bulkJointDeviation α f n p v ∂bulkParameterLaw) atTop (𝓝 0) := by |
| have h₁ := bulkDeviationAt_integral_tendsto (fun n => patternCount α (n+1)) f hα |
| (fun n => Nat.floor_le (mul_nonneg hα (Nat.cast_nonneg _))) p |
| have h₂ := omittedBulkDeviation_integral_tendsto hα f p |
| have he (n : ℕ) : (∫ v, bulkJointDeviation α f n p v ∂bulkParameterLaw) = |
| (∫ v, bulkDeviationAt n (patternCount α (n+1)) f p v ∂bulkParameterLaw) + |
| ∫ v, omittedBulkDeviation α f n p v ∂bulkParameterLaw := |
| integral_add (bulkDeviationAt_integrable _ _ _ _) (bulkDeviationAt_integrable _ _ _ _) |
| simp_rw [he] |
| simpa only [add_zero] using h₁.add h₂ |
|
|
| def bulkJointGoodSet (α : ℝ) (f : ℝ→ᵇℝ) (J : ℕ→ℕ) (n : ℕ) : Set (ℕ→ℝ) := |
| diagonalGoodSet bulkParameterLaw (bulkJointDeviation α f) J n ∩ {v | ∀ p, v p∈Icc (1:ℝ) 2} |
|
|
| lemma bulkJointGoodSet_measurable (α : ℝ) (f : ℝ→ᵇℝ) (J : ℕ→ℕ) (n : ℕ) : |
| MeasurableSet (bulkJointGoodSet α f J n) := by |
| refine (diagonalGoodSet_measurable _ _ (bulkJointDeviation_measurable α f) J n).inter ?_ |
| simp only [ofPred_forall] |
| exact MeasurableSet.iInter fun p => (measurable_pi_apply p) measurableSet_Icc |
|
|
| lemma bulkJointGoodSet_compl_measure (α : ℝ) (f : ℝ→ᵇℝ) (J : ℕ→ℕ) (n : ℕ) : |
| bulkParameterLaw.real (bulkJointGoodSet α f J n)ᶜ = |
| bulkParameterLaw.real (diagonalGoodSet bulkParameterLaw (bulkJointDeviation α f) J n)ᶜ := by |
| simp only [measureReal_def] |
| congr 1 |
| apply measure_congr |
| filter_upwards [bulkParameterLaw_ae] with v hv |
| apply propext |
| change ¬(v∈diagonalGoodSet bulkParameterLaw (bulkJointDeviation α f) J n ∧ ∀ p, v p∈Icc (1:ℝ) 2) ↔ _ |
| exact not_congr ⟨fun h => h.1, fun h => ⟨h,hv⟩⟩ |
|
|
| theorem exists_bulkJointGoodParameters {α : ℝ} (hα : 0≤α) (f : ℝ→ᵇℝ) : |
| ∃ J : ℕ→ℕ, Monotone J ∧ (∀ n, J n≤n) ∧ Tendsto J atTop atTop ∧ |
| (∀ n, MeasurableSet (bulkJointGoodSet α f J n)) ∧ |
| Tendsto (fun n => bulkParameterLaw.real (bulkJointGoodSet α f J n)ᶜ) atTop (𝓝 0) ∧ |
| (∀ v : ℕ→(ℕ→ℝ), (∀ᶠ n in atTop, v n∈bulkJointGoodSet α f J n) → |
| ∀ p, Tendsto (fun n => bulkDeviationAt n (patternCount α (n+1)) f p (v n)) atTop (𝓝 0) ∧ |
| Tendsto (fun n => omittedBulkDeviation α f n p (v n)) atTop (𝓝 0)) := by |
| have hn := bulkJointDeviation_nonneg α f |
| have hi := bulkJointDeviation_integrable α f |
| obtain ⟨J,hmon,hbound,hlim,hmass⟩ := exists_slow_diagonal |
| (fun n p => integral_nonneg (hn n p)) (bulkJointDeviation_integral_tendsto hα f) |
| refine ⟨J,hmon,hbound,hlim,bulkJointGoodSet_measurable α f J,?_,?_⟩ |
| · simpa only [bulkJointGoodSet_compl_measure] using |
| diagonalGoodSet_compl_tendsto bulkParameterLaw (bulkJointDeviation α f) hn hi J hmass |
| · intro v hv p |
| have hd := diagonalGoodSet_deviation_tendsto bulkParameterLaw (bulkJointDeviation α f) hn J hlim hmass v |
| (hv.mono fun n h => h.1) p |
| constructor |
| · exact squeeze_zero (fun n => bulkDeviationAt_nonneg _ _ _ _ _) |
| (fun n => le_add_of_nonneg_right (bulkDeviationAt_nonneg _ _ _ _ _)) hd |
| · exact squeeze_zero (fun n => bulkDeviationAt_nonneg _ _ _ _ _) |
| (fun n => le_add_of_nonneg_left (bulkDeviationAt_nonneg _ _ _ _ _)) hd |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
|
|
