| import Mathlib |
| import OAI.Probability.Perceptron.Variational.IntegratedConvex |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology BigOperators |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma euclideanReplica_measurable_left {I : Type} {S : Type*} [Fintype I] [MeasurableSpace S] |
| (μ : Measure S) [IsProbabilityMeasure μ] (r : ℕ) (j : Fin r) |
| {H : S→ℝ} {V W : S→EuclideanSpace ℝ I} {G : (Fin r→S)→ℝ} |
| (hH : Measurable H) (hV : Measurable V) (hW : Measurable W) (hG : Measurable G) : |
| Measurable (fun g => gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r |
| (fun x => inner ℝ (W (x j)) g*G x)) := by |
| have hh : Measurable (fun a : EuclideanSpace ℝ I × S => H a.2+inner ℝ (V a.2) a.1) := |
| (hH.comp measurable_snd).add ((hV.comp measurable_snd).inner measurable_fst) |
| have hmj : Measurable (fun x : Fin r→S => W (x j)) := hW.comp (measurable_pi_apply j) |
| have hl : Measurable (fun a : EuclideanSpace ℝ I × (Fin r→S) => inner ℝ (W (a.2 j)) a.1*G a.2) := |
| ((hmj.comp measurable_snd).inner measurable_fst).mul (hG.comp measurable_snd) |
| exact kernel_replicaMean_measurable (Kernel.const (EuclideanSpace ℝ I) μ) hh hl |
|
|
| lemma euclideanReplica_measurable_right {I : Type} {S : Type*} [Fintype I] [MeasurableSpace S] |
| (μ : Measure S) [IsProbabilityMeasure μ] (r : ℕ) (j : Fin r) |
| {H : S→ℝ} {V W : S→EuclideanSpace ℝ I} {G : (Fin r→S)→ℝ} |
| (hH : Measurable H) (hV : Measurable V) (hW : Measurable W) (hG : Measurable G) : |
| Measurable (fun g => gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r |
| (fun x => G x*∑ l, inner ℝ (W (x j)) (V (x l))) - |
| r*gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) (r+1) |
| (fun x => G (fun l => x l.succ)*inner ℝ (W (x j.succ)) (V (x 0)))) := by |
| have hh : Measurable (fun a : EuclideanSpace ℝ I × S => H a.2+inner ℝ (V a.2) a.1) := |
| (hH.comp measurable_snd).add ((hV.comp measurable_snd).inner measurable_fst) |
| have hsum : Measurable (fun x : Fin r→S => G x*∑ l, inner ℝ (W (x j)) (V (x l))) := by |
| apply hG.mul |
| exact Finset.measurable_sum _ fun l _ => (hW.comp (measurable_pi_apply j)).inner (hV.comp (measurable_pi_apply l)) |
| have hpair : Measurable (fun x : Fin (r+1)→S => G (fun l => x l.succ)*inner ℝ (W (x j.succ)) (V (x 0))) := |
| (hG.comp (Measurable.of_eval fun replica => measurable_pi_apply replica.succ)).mul |
| ((hW.comp (measurable_pi_apply j.succ)).inner (hV.comp (measurable_pi_apply 0))) |
| exact (kernel_replicaMean_measurable (Kernel.const (EuclideanSpace ℝ I) μ) |
| (H := fun g x => H x+inner ℝ (V x) g) |
| (G := fun _ x => G x*∑ l, inner ℝ (W (x j)) (V (x l))) hh |
| (hsum.comp measurable_snd)).sub ((kernel_replicaMean_measurable (Kernel.const (EuclideanSpace ℝ I) μ) |
| (H := fun g x => H x+inner ℝ (V x) g) |
| (G := fun _ x => G (fun l => x l.succ)*inner ℝ (W (x j.succ)) (V (x 0))) hh |
| (hpair.comp measurable_snd)).const_mul r) |
|
|
| lemma countableVector_replica_ibp {I : Type} {S : Type*} [Fintype I] [MeasurableSpace S] |
| (μ : Measure S) [IsProbabilityMeasure μ] (r : ℕ) (j : Fin r) |
| {H : S→ℝ} {V W : S→EuclideanSpace ℝ I} {G : (Fin r→S)→ℝ} |
| (hH : Measurable H) (hV : Measurable V) (hW : Measurable W) (hG : Measurable G) |
| {A D E B : ℝ} (hA : ∀ x, |H x|≤A) (hD : ∀ x, ‖V x‖^2≤D) |
| (hE : ∀ x, ‖W x‖^2≤E) (hB : 0≤B) (hGB : ∀ x, |G x|≤B) : |
| (∫ g, gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) (countableGaussianVector g)) r |
| (fun x => inner ℝ (W (x j)) (countableGaussianVector g)*G x) ∂countableGaussianLaw) = |
| ∫ g, gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) (countableGaussianVector g)) r |
| (fun x => G x*∑ l, inner ℝ (W (x j)) (V (x l))) - |
| r*gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) (countableGaussianVector g)) (r+1) |
| (fun x => G (fun l => x l.succ)*inner ℝ (W (x j.succ)) (V (x 0))) |
| ∂countableGaussianLaw := by |
| have he := countableGaussian_replica_full_ibp μ r j hH |
| (vectorGaussianRow_measurable hV) (vectorGaussianRow_measurable hW) measurable_const hG hA |
| (fun x => (vectorGaussianRow_norm V x).trans_le (hD x)) |
| (fun x => (vectorGaussianRow_norm W x).trans_le (hE x)) hB hGB |
| have hham (g) : countableGaussianHamiltonian H (vectorGaussianRow V) |
| (fun _ => vectorGaussianLength I) g=fun x => H x+inner ℝ (V x) (countableGaussianVector g) := by |
| funext x |
| exact congrArg (fun y => H x+y) (vectorGaussianRow_field V g x) |
| simp_rw [hham,vectorGaussianRow_field,vectorGaussianRow_covariance] at he |
| exact he |
|
|
| lemma euclideanGaussian_replica_ibp {I : Type} {S : Type*} [Fintype I] [MeasurableSpace S] |
| (μ : Measure S) [IsProbabilityMeasure μ] (r : ℕ) (j : Fin r) |
| {H : S→ℝ} {V W : S→EuclideanSpace ℝ I} {G : (Fin r→S)→ℝ} |
| (hH : Measurable H) (hV : Measurable V) (hW : Measurable W) (hG : Measurable G) |
| {A D E B : ℝ} (hA : ∀ x, |H x|≤A) (hD : ∀ x, ‖V x‖^2≤D) |
| (hE : ∀ x, ‖W x‖^2≤E) (hB : 0≤B) (hGB : ∀ x, |G x|≤B) : |
| (∫ g, gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r |
| (fun x => inner ℝ (W (x j)) g*G x) ∂stdGaussian (EuclideanSpace ℝ I)) = |
| ∫ g, gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r |
| (fun x => G x*∑ l, inner ℝ (W (x j)) (V (x l))) - |
| r*gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) (r+1) |
| (fun x => G (fun l => x l.succ)*inner ℝ (W (x j.succ)) (V (x 0))) |
| ∂stdGaussian (EuclideanSpace ℝ I) := by |
| have hleft := euclideanReplica_measurable_left μ r j hH hV hW hG |
| have hright := euclideanReplica_measurable_right μ r j hH hV hW hG |
| rw [← countableGaussianVector_law (I:=I), |
| integral_map countableGaussianVector_measurable.aemeasurable hleft.aestronglyMeasurable, |
| integral_map countableGaussianVector_measurable.aemeasurable hright.aestronglyMeasurable] |
| exact countableVector_replica_ibp μ r j hH hV hW hG hA hD hE hB hGB |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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