| import Mathlib |
| import OAI.Probability.Perceptron.Variational.Contact |
| import OAI.Probability.Perceptron.Interpolation.KernelReplicaGibbs |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def sourceCouplingCovariance (n k : ℕ) (p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) |
| (j : Fin (n+1)) (x y : NormalizedSpin (n+1)×IndexedLeaf k) := |
| perturbationAmplitude (n+1) (fun _ => 1) j * perturbationAmplitude (n+1) u j * |
| sourceJointMonomial p d j x y |
|
|
| lemma sourceCouplingCovariance_measurable (n k r : ℕ) (p d : Fin (n+1) → ℕ) |
| (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (i l : Fin r) : |
| Measurable (fun x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k => |
| sourceCouplingCovariance n k p d u j (x i) (x l)) := by |
| have hh := (sourceReplica_covariance_measurable (N := n+1) (k := k) p d j i l).const_mul |
| (perturbationAmplitude (n+1) (fun _ => 1) j * perturbationAmplitude (n+1) u j) |
| exact hh |
|
|
| lemma sourceCouplingCovariance_bound (n k : ℕ) (p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) |
| (j : Fin (n+1)) (x y : NormalizedSpin (n+1)×IndexedLeaf k) : |
| |sourceCouplingCovariance n k p d u j x y| ≤ |
| |perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j| := by |
| unfold sourceCouplingCovariance |
| rw [abs_mul] |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left |
| (sourceJointMonomial_abs_le p d j x y) (abs_nonneg (perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j)) |
|
|
| lemma sourceCoupling_conditional_ibp (n k r : ℕ) (f : ℝ →ᵇ ℝ) |
| (p d : Fin (n+1) → ℕ) (h : Fin (k+1) → ℝ) |
| (hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) (u : Fin (n+1) → ℝ) |
| (j : Fin (n+1)) (i : Fin r) (b : SourceBaseData n k) |
| {G : (Fin r → NormalizedSpin (n+1)×IndexedLeaf k) → ℝ} |
| (hG : Measurable G) {B : ℝ} (hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B) : |
| (∫ g, gibbsReplicaMean (sourceSpinLeafKernel n k b) |
| (sourceCouplingHamiltonian n k f p d h u (b,g)) r |
| (fun x => sourceCouplingEnergy n k p d h j (b,g) (x i)*G x) ∂countableGaussianLaw) = |
| ∫ g, gibbsReplicaMean (sourceSpinLeafKernel n k b) |
| (sourceCouplingHamiltonian n k f p d h u (b,g)) r |
| (fun x => G x*∑ l, sourceCouplingCovariance n k p d u j (x i) (x l)) - |
| r*gibbsReplicaMean (sourceSpinLeafKernel n k b) |
| (sourceCouplingHamiltonian n k f p d h u (b,g)) (r+1) |
| (fun x => G (fun l => x l.succ)*sourceCouplingCovariance n k p d u j (x i.succ) (x 0)) |
| ∂countableGaussianLaw := by |
| let c := perturbationAmplitude (n+1) (fun _ => 1) j |
| have hb := sourceReplica_gaussian_ibp n b.1 k r f (patternPrefix (n+1) b.1 b.2.1) |
| p d h hh0 hh u b.2.2 j i hG hB hGB |
| dsimp only at hb |
| have he := congrArg (fun x : ℝ => c*x) hb |
| simp only [sourceSpinLeafKernel_apply,sourceCouplingHamiltonian,sourceCouplingEnergy, |
| sourceCouplingCovariance] |
| have hs (x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k) : |
| G x*∑ l, c*perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l) = |
| c*(G x*∑ l, perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l)) := by |
| simp only [mul_assoc,Finset.mul_sum] |
| ring_nf |
| simp only [mul_assoc] |
| change (∫ g, gibbsReplicaMean (enrichedIndexedBaseMeasure n k b.2.2) |
| (enrichedIndexedHamiltonian n b.1 k f (patternPrefix (n+1) b.1 b.2.1) p d h u g) r |
| (fun x => c*(countableGaussianField (sourceGaussianTestRow p d h j) |
| (indexedGaussianRowLength (I := EnrichedIndex (n+1) (n+1) p) k) g (x i)*G x)) ∂countableGaussianLaw) = _ |
| simp_rw [replicaMean_const_mul,integral_const_mul] |
| rw [hb, ← integral_const_mul] |
| apply integral_congr_ae |
| exact ae_of_all _ fun g => by |
| have hs₁ : (fun x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k => |
| G x * ∑ l, c*(perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l))) = |
| (fun x => c*(G x*∑ l, perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l))) := by |
| funext x |
| rw [← Finset.mul_sum] |
| ring |
| have hs₂ : (fun x : Fin (r+1) → NormalizedSpin (n+1)×IndexedLeaf k => |
| G (fun l => x l.succ)*(c*(perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i.succ) (x 0)))) = |
| (fun x => c*(G (fun l => x l.succ)*(perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i.succ) (x 0)))) := by |
| funext x |
| ring |
| rw [hs₁, hs₂] |
| simp only [replicaMean_const_mul] |
| ring |
|
|
| lemma sourceCoupling_annealed_ibp (n k r : ℕ) (f : ℝ →ᵇ ℝ) |
| (p d : Fin (n+1) → ℕ) (h : Fin (k+1) → ℝ) |
| (hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) (u : Fin (n+1) → ℝ) |
| (j : Fin (n+1)) (i : Fin r) (z : Fin k → ℝ) (t : ℝ≥0) |
| {G : (Fin r → NormalizedSpin (n+1)×IndexedLeaf k) → ℝ} |
| (hG : Measurable G) {B : ℝ} (hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B) |
| (hYi : Integrable (fun a => tiltMean (sourceSpinLeafKernel n k a.1) |
| (sourceCouplingHamiltonian n k f p d h u a) |
| (fun x => |sourceCouplingEnergy n k p d h j a x|) 1) |
| ((sourceBaseDataLaw n k z t).prod countableGaussianLaw)) : |
| let H := sourceCouplingHamiltonian n k f p d h u |
| let Y := sourceCouplingEnergy n k p d h j |
| let C := sourceCouplingCovariance n k p d u j |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| (∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) r (fun x => Y a (x i)*G x) ∂P) = |
| ∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) r (fun x => G x*∑ l, C (x i) (x l)) - |
| r*gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) (r+1) |
| (fun x => G (fun l => x l.succ)*C (x i.succ) (x 0)) ∂P := by |
| dsimp only |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| let H := sourceCouplingHamiltonian n k f p d h u |
| let Y := sourceCouplingEnergy n k p d h j |
| let C := sourceCouplingCovariance n k p d u j |
| let C₀ := |perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j| |
| have hH := sourceCouplingHamiltonian_measurable n k f p d h u |
| have hY := sourceCouplingEnergy_measurable n k p d h j |
| have he := sourceCoupling_all_exp_ae n k f p d h hh0 hh u j z t |
| have he0 : ∀ᵐ a ∂P, Integrable (fun x => Real.exp (H a x)) (sourceSpinLeafKernel n k a.1) := by |
| filter_upwards [he] with a ha |
| simpa only [zero_mul,add_zero] using ha 0 |
| have he1 : ∀ᵐ a ∂P, Integrable (fun x => Real.exp (H a x)*Y a x) (sourceSpinLeafKernel n k a.1) := by |
| filter_upwards [he] with a ha |
| simpa only [zero_mul,add_zero,pow_one] using coupling_weighted_power_integrable |
| (sourceSpinLeafKernel n k a.1) hH.of_uncurry_left hY.of_uncurry_left ha 0 1 |
| have hi := kernel_replica_energy_integrable (sourceFullSpinLeafKernel n k) P i hH hY hG he0 he1 hYi hB hGB |
| have hmG : Measurable (fun x : Fin (r+1) → NormalizedSpin (n+1)×IndexedLeaf k => G (fun l => x l.succ)) := |
| hG.comp (Measurable.of_eval fun coordinate => measurable_pi_apply coordinate.succ) |
| have hsum : ∀ x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k, |∑ l, C (x i) (x l)| ≤ r*C₀ := by |
| intro x |
| apply (Finset.abs_sum_le_sum_abs _ _).trans |
| calc |
| _ ≤ ∑ _ : Fin r, C₀ := Finset.sum_le_sum fun l _ => sourceCouplingCovariance_bound n k p d u j _ _ |
| _ = _ := by simp |
| have h1G : Measurable (fun x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k => G x*∑ l, C (x i) (x l)) := |
| hG.mul (Finset.measurable_sum _ fun l _ => sourceCouplingCovariance_measurable n k r p d u j i l) |
| have h2G : Measurable (fun x : Fin (r+1) → NormalizedSpin (n+1)×IndexedLeaf k => G (fun l => x l.succ)*C (x i.succ) (x 0)) := |
| hmG.mul (sourceCouplingCovariance_measurable n k (r+1) p d u j i.succ 0) |
| have hi1 := kernel_replicaMean_bounded_integrable (sourceFullSpinLeafKernel n k) P |
| (G := fun _ x => G x*∑ l, C (x i) (x l)) hH (h1G.comp measurable_snd) |
| (show 0 ≤ B*(r*C₀) by positivity) (fun _ x => by |
| rw [abs_mul]; exact mul_le_mul (hGB x) (hsum x) (abs_nonneg _) hB) |
| have hi2 := kernel_replicaMean_bounded_integrable (sourceFullSpinLeafKernel n k) P |
| (G := fun _ x => G (fun l => x l.succ)*C (x i.succ) (x 0)) hH (h2G.comp measurable_snd) |
| (show 0 ≤ B*C₀ by positivity) (fun _ x => by |
| rw [abs_mul]; exact mul_le_mul (hGB _) (sourceCouplingCovariance_bound n k p d u j _ _) (abs_nonneg _) hB) |
| change (∫ a, gibbsReplicaMean (sourceFullSpinLeafKernel n k a) (H a) r (fun x => Y a (x i)*G x) ∂P) = |
| ∫ a, gibbsReplicaMean (sourceFullSpinLeafKernel n k a) (H a) r (fun x => G x*∑ l, C (x i) (x l))- |
| r*gibbsReplicaMean (sourceFullSpinLeafKernel n k a) (H a) (r+1) (fun x => G (fun l => x l.succ)*C (x i.succ) (x 0)) ∂P |
| have hiR := hi1.sub (hi2.const_mul (r:ℝ)) |
| dsimp only [sourceFullSpinLeafKernel,Kernel.comap_apply,P,H,Y,C,Pi.sub_apply] at hi hiR ⊢ |
| have hprodR := integral_prod _ hiR |
| simp only [Pi.sub_apply] at hprodR |
| rw [integral_prod _ hi, hprodR] |
| apply integral_congr_ae |
| exact ae_of_all _ fun b => sourceCoupling_conditional_ibp n k r f p d h hh0 hh u j i b hG hB hGB |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
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|
