openAI-math-dataset / lean /OAI /Probability /Perceptron /Interpolation /CountableGaussianReplicaIBP.lean
| import Mathlib | |
| import OAI.Probability.Perceptron.Pressure.GaussianHamiltonianCoordinates | |
| import OAI.Probability.Perceptron.Variational.CountableGaussianIBP | |
| namespace OAI | |
| noncomputable section | |
| namespace SphericalPerceptronFreeEnergy | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology NNReal ENNReal BigOperators | |
| variable {S : Type*} [MeasurableSpace S] (μ : Measure S) [IsProbabilityMeasure μ] | |
| lemma countableGaussian_replica_coordinate_ibp (i n : ℕ) | |
| {H : (ℕ → ℝ) → S → ℝ} {v : S → ℝ} {G : (Fin n → S) → ℝ} | |
| (hH : Measurable (Function.uncurry H)) (hv : Measurable v) (hG : Measurable G) | |
| {C D : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D) | |
| (hvC : ∀ x, |v x| ≤ C) (hGD : ∀ x, |G x| ≤ D) | |
| (haff : ∀ g t x, H (Function.update g i t) x = H (Function.update g i 0) x+t*v x) | |
| (hi : ∀ᵐ g ∂countableGaussianLaw, Integrable (fun x => Real.exp (H g x)) μ) : | |
| (∫ g, g i*gibbsReplicaMean μ (H g) n G ∂countableGaussianLaw) = | |
| ∫ g, gibbsReplicaMean μ (H g) n (fun x => replicaPotential v n x*G x)- | |
| n*gibbsReplicaMean μ (H g) (n+1) (fun x => v (x 0)*G (fun l => x l.succ)) | |
| ∂countableGaussianLaw := by | |
| let Hr (g : ℕ → ℝ) := replicaPotential (H g) n | |
| have hHr : Measurable (Function.uncurry Hr) := by | |
| change Measurable (fun p : (ℕ → ℝ)×(Fin n → S) => ∑ l, H p.1 (p.2 l)) | |
| exact Finset.measurable_sum _ fun l _ => hH.comp | |
| (measurable_fst.prodMk ((measurable_pi_apply l).comp measurable_snd)) | |
| have hvr := replicaPotential_measurable hv n | |
| have hir : ∀ᵐ g ∂countableGaussianLaw, Integrable (fun x => Real.exp (Hr g x)) | |
| (Measure.pi fun _ : Fin n => μ) := by | |
| filter_upwards [hi] with g hg | |
| simpa only [one_mul] using replica_exp_integrable μ (by simpa only [one_mul] using hg : | |
| Integrable (fun x => Real.exp (1*H g x)) μ) n | |
| have haffr (g : ℕ → ℝ) (t : ℝ) (x : Fin n → S) : | |
| Hr (Function.update g i t) x = Hr (Function.update g i 0) x+t*replicaPotential v n x := by | |
| calc | |
| _ = ∑ l, (H (Function.update g i 0) (x l)+t*v (x l)) := | |
| Finset.sum_congr rfl (fun l _ => haff g t (x l)) | |
| _ = _ := by simp only [Hr,replicaPotential,Finset.sum_add_distrib,Finset.mul_sum] | |
| have hh := countableGaussian_coordinate_tilt_ibp_of_exp_integrable | |
| (Measure.pi fun _ : Fin n => μ) i hHr hvr hG (mul_nonneg (Nat.cast_nonneg n) hC) hD | |
| (replicaPotential_bound hvC n) hGD haffr hir | |
| change (∫ g, g i*gibbsReplicaMean μ (H g) n G ∂countableGaussianLaw) = | |
| ∫ g, gibbsReplicaMean μ (H g) n (fun x => replicaPotential v n x*G x)- | |
| gibbsReplicaMean μ (H g) n G*gibbsReplicaMean μ (H g) n (replicaPotential v n) | |
| ∂countableGaussianLaw at hh | |
| refine hh.trans ?_ | |
| apply integral_congr_ae | |
| filter_upwards [hi] with g hg | |
| have hHg : Measurable (H g) := hH.comp (measurable_const.prodMk measurable_id) | |
| rw [gibbsReplicaMean_potential_of_integrable μ hHg hv hg hvC, | |
| gibbsReplicaMean_product_of_integrable μ hHg hv hG hg hvC hGD] | |
| ring | |
| lemma gaussianHamiltonian_replica_coordinate_ibp (i n : ℕ) | |
| {W : S → ℝ} {v : ℕ → S → ℝ} {L : S → ℕ} {G : (Fin n → S) → ℝ} | |
| (hW : Measurable W) (hv : ∀ j, Measurable (v j)) (hL : Measurable L) (hG : Measurable G) | |
| {A D B : ℝ} (hA : ∀ x, |W x| ≤ A) | |
| (hD : ∀ x, (∑ j : Fin (L x), v j.val x^2) ≤ D) | |
| (hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B) : | |
| let H := countableGaussianHamiltonian W v L | |
| let a := maskedGaussianCoefficient v L i | |
| (∫ g, g i*gibbsReplicaMean μ (H g) n G ∂countableGaussianLaw) = | |
| ∫ g, gibbsReplicaMean μ (H g) n (fun x => replicaPotential a n x*G x)- | |
| n*gibbsReplicaMean μ (H g) (n+1) (fun x => a (x 0)*G (fun l => x l.succ)) | |
| ∂countableGaussianLaw := by | |
| dsimp only | |
| apply countableGaussian_replica_coordinate_ibp μ i n | |
| (countableGaussianHamiltonian_measurable hW hv hL) | |
| (maskedGaussianCoefficient_measurable hv hL i) hG (Real.sqrt_nonneg D) hB | |
| (maskedGaussianCoefficient_bound v L hD i) hGB | |
| (fun g t x => countableGaussianHamiltonian_update W v L g i t x) | |
| simpa only [one_mul] using | |
| (countableGaussianHamiltonian_exp_joint_integrable μ hW hv hL hA hD 1).prod_left_ae | |
| lemma integrable_tiltLaw_of_weighted {H F : S → ℝ} (hH : Measurable H) | |
| (he : Integrable (fun x => Real.exp (H x)) μ) | |
| (hF : Integrable (fun x => Real.exp (H x)*F x) μ) : Integrable F (tiltLaw μ H 1) := by | |
| have hp := tilt_partition_pos_of_integrable μ (by simpa only [one_mul] using he : | |
| Integrable (fun x => Real.exp (1*H x)) μ) | |
| unfold tiltLaw | |
| apply Integrable.smul_measure _ (ENNReal.inv_ne_top.mpr (ne_of_gt (ENNReal.ofReal_pos.mpr hp))) | |
| apply (integrable_withDensity_iff_integrable_smul' (hH.const_mul 1).exp.ennreal_ofReal | |
| (ae_of_all _ fun _ => ENNReal.ofReal_lt_top)).mpr | |
| simpa only [one_mul,ENNReal.toReal_ofReal (Real.exp_pos _).le,smul_eq_mul] using hF | |
| lemma replicaMean_varying_measurable {Ω : Type*} [MeasurableSpace Ω] {H : Ω → S → ℝ} | |
| {n : ℕ} {F : Ω → (Fin n → S) → ℝ} (hH : Measurable (Function.uncurry H)) | |
| (hF : Measurable (Function.uncurry F)) : | |
| Measurable (fun g => gibbsReplicaMean μ (H g) n (F g)) := by | |
| have hr : Measurable (fun p : Ω×(Fin n → S) => Real.exp (replicaPotential (H p.1) n p.2)) := by | |
| apply Measurable.exp | |
| exact Finset.measurable_sum _ fun i _ => hH.comp | |
| (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd)) | |
| let ρ := Measure.pi fun _ : Fin n => μ | |
| have hn : Measurable (fun g => ∫ x, Real.exp (replicaPotential (H g) n x)*F g x ∂ρ) := | |
| (hr.mul hF).stronglyMeasurable.integral_prod_right'.measurable | |
| have hd : Measurable (fun g => ∫ x, Real.exp (replicaPotential (H g) n x) ∂ρ) := | |
| hr.stronglyMeasurable.integral_prod_right'.measurable | |
| unfold gibbsReplicaMean tiltMean tiltIntegral tiltPartition | |
| simp only [one_mul] | |
| convert hn.div hd using 1 | |
| lemma replicaMean_energy_bound {H F : S → ℝ} {n : ℕ} (j : Fin n) | |
| {G : (Fin n → S) → ℝ} (hH : Measurable H) (hF : Measurable F) (hG : Measurable G) | |
| (he : Integrable (fun x => Real.exp (H x)) μ) | |
| (hFint : Integrable (fun x => Real.exp (H x)*F x) μ) | |
| {B : ℝ} (hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B) : | |
| |gibbsReplicaMean μ H n (fun x => F (x j)*G x)| ≤ B*tiltMean μ H (fun x => |F x|) 1 := by | |
| have he1 : Integrable (fun x => Real.exp (1*H x)) μ := by simpa only [one_mul] using he | |
| let := tilt_law_probability_of_integrable μ he1 | |
| let θ := tiltLaw μ H 1 | |
| let ρ := Measure.pi fun _ : Fin n => θ | |
| have hFi : Integrable F θ := integrable_tiltLaw_of_weighted μ hH he hFint | |
| have hcoord : Integrable (fun x : Fin n → S => F (x j)) ρ := | |
| (measurePreserving_eval (fun _ : Fin n => θ) j).integrable_comp_of_integrable hFi | |
| have hprod : Integrable (fun x : Fin n → S => F (x j)*G x) ρ := by | |
| apply (hcoord.abs.const_mul B).mono' ((hF.comp (measurable_pi_apply j)).mul hG).aestronglyMeasurable | |
| exact ae_of_all _ fun x => by | |
| change |F (x j)*G x| ≤ B*|F (x j)| | |
| rw [abs_mul,mul_comm |F (x j)|] | |
| exact mul_le_mul (hGB x) le_rfl (abs_nonneg _) hB | |
| rw [gibbsReplicaMean_integral_of_integrable μ hH he] | |
| calc | |
| _ ≤ ∫ x : Fin n → S, |F (x j)*G x| ∂ρ := by | |
| simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := ρ) (fun x : Fin n → S => F (x j)*G x) | |
| _ ≤ ∫ x : Fin n → S, B*|F (x j)| ∂ρ := by | |
| apply integral_mono hprod.abs (hcoord.abs.const_mul B) | |
| intro x | |
| change |F (x j)*G x| ≤ B*|F (x j)| | |
| rw [abs_mul] | |
| nlinarith [mul_le_mul_of_nonneg_left (hGB x) (abs_nonneg (F (x j)))] | |
| _ = _ := by | |
| rw [integral_const_mul] | |
| congr 1 | |
| rw [← gibbsReplicaMean_coordinate_of_integrable μ hH hF.abs he n j, | |
| gibbsReplicaMean_integral_of_integrable μ hH he] | |
| lemma replicaMean_energy_sub {H F Q : S → ℝ} {n : ℕ} (j : Fin n) | |
| {G : (Fin n → S) → ℝ} (hH : Measurable H) (hF : Measurable F) (hQ : Measurable Q) (hG : Measurable G) | |
| (he : Integrable (fun x => Real.exp (H x)) μ) | |
| (hFint : Integrable (fun x => Real.exp (H x)*F x) μ) | |
| (hQint : Integrable (fun x => Real.exp (H x)*Q x) μ) | |
| {B : ℝ} (hGB : ∀ x, |G x| ≤ B) : | |
| gibbsReplicaMean μ H n (fun x => (F (x j)-Q (x j))*G x) = | |
| gibbsReplicaMean μ H n (fun x => F (x j)*G x)-gibbsReplicaMean μ H n (fun x => Q (x j)*G x) := by | |
| have he1 : Integrable (fun x => Real.exp (1*H x)) μ := by simpa only [one_mul] using he | |
| let := tilt_law_probability_of_integrable μ he1 | |
| let θ := tiltLaw μ H 1 | |
| let ρ := Measure.pi fun _ : Fin n => θ | |
| have him (f : S → ℝ) (hf : Measurable f) (hfi : Integrable (fun x => Real.exp (H x)*f x) μ) : | |
| Integrable (fun x : Fin n → S => f (x j)*G x) ρ := by | |
| have hc := (measurePreserving_eval (fun _ : Fin n => θ) j).integrable_comp_of_integrable | |
| (integrable_tiltLaw_of_weighted μ hH he hfi) | |
| apply (hc.abs.const_mul B).mono' ((hf.comp (measurable_pi_apply j)).mul hG).aestronglyMeasurable | |
| exact ae_of_all _ fun x => by | |
| change |f (x j)*G x| ≤ B*|f (x j)| | |
| rw [abs_mul] | |
| nlinarith [mul_le_mul_of_nonneg_left (hGB x) (abs_nonneg (f (x j)))] | |
| simp_rw [gibbsReplicaMean_integral_of_integrable μ hH he,sub_mul] | |
| exact integral_sub (him F hF hFint) (him Q hQ hQint) | |
| end SphericalPerceptronFreeEnergy | |
| end | |
| end OAI | |