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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import Mathlib
import OAI.Probability.Perceptron.Interpolation.AnnealedReplicaConvergence
import OAI.Probability.Perceptron.Variational.GaussianGibbsEnergy
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SphericalPerceptronFreeEnergy
variable {S : Type*} [MeasurableSpace S] (μ : Measure S) [IsProbabilityMeasure μ]
lemma countableGaussian_replica_full_ibp (n : ℕ) (j : Fin n)
{W : S → ℝ} {v w : ℕ → S → ℝ} {L : S → ℕ} {G : (Fin n → S) → ℝ}
(hW : Measurable W) (hv : ∀ i, Measurable (v i)) (hw : ∀ i, Measurable (w i))
(hL : Measurable L) (hG : Measurable G) {A D E B : ℝ}
(hA : ∀ x, |W x| ≤ A) (hD : ∀ x, (∑ i : Fin (L x), v i.val x^2) ≤ D)
(hE : ∀ x, (∑ i : Fin (L x), w i.val x^2) ≤ E)
(hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B) :
let H := countableGaussianHamiltonian W v L
(∫ g, gibbsReplicaMean μ (H g) n (fun x => countableGaussianField w L g (x j)*G x)
∂countableGaussianLaw) =
∫ g, gibbsReplicaMean μ (H g) n (fun x => G x*∑ l, countableGaussianCovariance v w L (x j) (x l)) -
n*gibbsReplicaMean μ (H g) (n+1)
(fun x => G (fun l => x l.succ)*countableGaussianCovariance v w L (x j.succ) (x 0)) ∂countableGaussianLaw := by
dsimp only
let H := countableGaussianHamiltonian W v L
let C := |(D+E)/2|
have hC : 0 ≤ C := abs_nonneg _
have hHm : Measurable (Function.uncurry H) := countableGaussianHamiltonian_measurable hW hv hL
have he : ∀ᵐ g ∂countableGaussianLaw, Integrable (fun x => Real.exp (H g x)) μ := by
simpa only [one_mul] using (countableGaussianHamiltonian_exp_joint_integrable μ hW hv hL hA hD 1).prod_left_ae
let K := gaussianPrefixCovariance v w L
let KFull := countableGaussianCovariance v w L
have hKm (m) : Measurable (Function.uncurry (K m)) := gaussianPrefixCovariance_measurable hv hw hL m
have hKFullm : Measurable (Function.uncurry KFull) := countableGaussianCovariance_measurable hv hw hL
have hKb (m x y) : |K m x y| ≤ C := (gaussianPrefixCovariance_bound v w L hD hE m x y).trans (le_abs_self _)
have hKFullb (x y) : |KFull x y| ≤ C := hKb (L x) x y
have hKt (x y) : Tendsto (fun m => K m x y) atTop (nhds (KFull x y)) :=
tendsto_const_nhds.congr' ((gaussianPrefixCovariance_eventually_eq v w L x y).mono fun _ h => h.symm)
let U (m) (x : Fin n → S) := G x*∑ l, K m (x j) (x l)
let UFull (x : Fin n → S) := G x*∑ l, KFull (x j) (x l)
let V (m) (x : Fin (n+1) → S) := G (fun l => x l.succ)*K m (x j.succ) (x 0)
let VFull (x : Fin (n+1) → S) := G (fun l => x l.succ)*KFull (x j.succ) (x 0)
have hPair (a b : Fin n) : Measurable (fun x : Fin n → S => (x a,x b)) :=
(measurable_pi_apply a).prodMk (measurable_pi_apply b)
have hUm (m) : Measurable (U m) := hG.mul
(Finset.measurable_sum Finset.univ fun (l : Fin n) _ => (hKm m).comp (hPair j l))
have hUFullm : Measurable UFull := hG.mul
(Finset.measurable_sum Finset.univ fun (l : Fin n) _ => hKFullm.comp (hPair j l))
have hPairS : Measurable (fun x : Fin (n+1) → S => (x j.succ,x 0)) :=
(measurable_pi_apply j.succ).prodMk (measurable_pi_apply 0)
have hVm (m) : Measurable (V m) :=
(hG.comp (Measurable.of_eval fun index : Fin n => measurable_pi_apply index.succ)).mul
((hKm m).comp hPairS)
have hVFullm : Measurable VFull :=
(hG.comp (Measurable.of_eval fun index : Fin n => measurable_pi_apply index.succ)).mul
(hKFullm.comp hPairS)
have hsum (k : S → S → ℝ) (hk : ∀ x y, |k x y| ≤ C) (x : Fin n → S) :
|G x*∑ l, k (x j) (x l)| ≤ B*((n:ℝ)*C) := by
rw [abs_mul]
apply mul_le_mul (hGB x) _ (abs_nonneg _) hB
calc
_ ≤ ∑ l : Fin n, |k (x j) (x l)| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ l : Fin n, C := Finset.sum_le_sum fun l _ => hk _ _
_ = _ := by simp
have hUB (m x) : |U m x| ≤ B*((n:ℝ)*C) := hsum (K m) (hKb m) x
have hUFullB (x) : |UFull x| ≤ B*((n:ℝ)*C) := hsum KFull hKFullb x
have hVB (m x) : |V m x| ≤ B*C := by
exact (abs_mul _ _).trans_le (mul_le_mul (hGB _) (hKb m _ _) (abs_nonneg _) hB)
have hVFullB (x) : |VFull x| ≤ B*C := by
exact (abs_mul _ _).trans_le (mul_le_mul (hGB _) (hKFullb _ _) (abs_nonneg _) hB)
have hUt (x) : Tendsto (fun m => U m x) atTop (nhds (UFull x)) :=
(tendsto_finsetSum _ (fun index _ => hKt (x j) (x index))).const_mul (G x)
have hVt (x) : Tendsto (fun m => V m x) atTop (nhds (VFull x)) :=
(hKt (x j.succ) (x 0)).const_mul (G (fun l => x l.succ))
have hUi (m) := annealedReplica_bounded_integrable μ countableGaussianLaw hHm (hUm m) (by positivity) (hUB m)
have hVi (m) := annealedReplica_bounded_integrable μ countableGaussianLaw hHm (hVm m) (by positivity) (hVB m)
have hUFulli := annealedReplica_bounded_integrable μ countableGaussianLaw hHm hUFullm (by positivity) hUFullB
have hVFulli := annealedReplica_bounded_integrable μ countableGaussianLaw hHm hVFullm (by positivity) hVFullB
have hUL := annealedReplica_bounded_tendsto μ countableGaussianLaw hHm hUm hUFullm (by positivity) hUB hUt he
have hVL := annealedReplica_bounded_tendsto μ countableGaussianLaw hHm hVm hVFullm (by positivity) hVB hVt he
have hR := hUL.sub (hVL.const_mul (n:ℝ))
have hL' := gaussianEnergy_replica_prefix_tendsto μ hW hv hw hL hA hD hE j hG hB hGB
have heq (m) : (∫ g, gibbsReplicaMean μ (H g) n
(fun x => truncatedCountableGaussianField w L m g (x j)*G x) ∂countableGaussianLaw) =
(∫ g, gibbsReplicaMean μ (H g) n (U m) ∂countableGaussianLaw)-
(n:ℝ)*(∫ g, gibbsReplicaMean μ (H g) (n+1) (V m) ∂countableGaussianLaw) := by
rw [gaussianReplica_prefix_ibp μ m n j hW hv hw hL hG hA hD hE hB hGB]
exact (integral_sub (hUi m) ((hVi m).const_mul n)).trans (by rw [integral_const_mul])
have hlim := tendsto_nhds_unique hL' (hR.congr (fun m => (heq m).symm))
exact hlim.trans ((integral_sub hUFulli (hVFulli.const_mul n)).trans (by rw [integral_const_mul])).symm
end SphericalPerceptronFreeEnergy
end
end OAI
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