| import Mathlib |
| import OAI.Probability.Perceptron.Variational.FieldDerivative |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal NNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def sourceFieldPolynomial {k : ℕ} (a : Fin (k+1)→ℝ) : Polynomial ℝ := |
| Lagrange.interpolate Finset.univ (fun l : Fin (k+1) => (l.val:ℝ)/(k+1:ℕ)) a |
|
|
| lemma sourceFieldPolynomial_eval {k : ℕ} (a : Fin (k+1)→ℝ) (l : Fin (k+1)) : |
| (sourceFieldPolynomial a).eval ((l.val:ℝ)/(k+1:ℕ))=a l := by |
| unfold sourceFieldPolynomial |
| apply Lagrange.eval_interpolate_at_node (s := Finset.univ) |
| (v := fun l : Fin (k+1) => (l.val:ℝ)/(k+1:ℕ)) (i := l) a |
| · intro i hi j hj he |
| apply Fin.ext |
| have hv : (i.val:ℝ)=(j.val:ℝ) := (div_left_inj' (by positivity : ((k+1:ℕ):ℝ)≠0)).mp he |
| exact_mod_cast hv |
| · exact Finset.mem_univ l |
|
|
| lemma sourceFieldPolynomial_continuous {k : ℕ} (a : Fin (k+1)→ℝ) : |
| Continuous (fun t : unitInterval => (sourceFieldPolynomial a).eval t.val) := |
| (sourceFieldPolynomial a).continuous.comp continuous_subtype_val |
|
|
| def sourceFieldArrayTest {k : ℕ} (a : Fin (k+1)→ℝ) : CompactArray CompactJointOverlap→ᵇ ℝ := |
| BoundedContinuousFunction.mkOfCompact ⟨fun Q => (sourceFieldPolynomial a).eval (Q 1 0).2.val*(Q 1 0).1.val, |
| by fun_prop⟩ |
|
|
| lemma sourceGibbsArray_field_integral (n k : 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) (z : Fin k→ℝ) (t : ℝ≥0) (a : Fin (k+1)→ℝ) : |
| (∫ Q, sourceFieldArrayTest a Q ∂sourceGibbsArrayLaw n k f p d h u z t)= |
| ∫ b, gibbsReplicaMean (sourceSpinLeafKernel n k b.1) |
| (sourceCouplingHamiltonian n k f p d h u b) 2 (sourceFieldObservable a) |
| ∂((sourceBaseDataLaw n k z t).prod countableGaussianLaw) := by |
| let F : CompactBlock CompactJointOverlap 2→ᵇ ℝ := BoundedContinuousFunction.mkOfCompact |
| ⟨fun Q => (sourceFieldPolynomial a).eval (Q 1 0).2.val*(Q 1 0).1.val,by fun_prop⟩ |
| have he := sourceGibbsArray_block_integral n k f p d h hh0 hh u z t 2 F.measurable |
| (fun Q => F.norm_coe_le_norm Q) |
| change (∫ Q, sourceFieldArrayTest a Q ∂sourceGibbsArrayLaw n k f p d h u z t)=_ at he |
| rw [he] |
| apply integral_congr_ae |
| exact ae_of_all _ fun b => by |
| change gibbsReplicaMean (sourceSpinLeafKernel n k b.1) _ 2 _ = _ |
| congr 1 |
| funext x |
| dsimp [F,sourceJointOverlap,sourceFieldObservable] |
| rw [sourceFieldPolynomial_eval] |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
|
|
