| import Mathlib |
| import OAI.Probability.Perceptron.Pressure.SphericalPatternDirection |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma spherical_pattern_radial_ibp {n : ℕ} |
| (μ : Measure (NormalizedSpin (n+1))) [IsProbabilityMeasure μ] |
| (f : Jet3) {h : NormalizedSpin (n+1)→ℝ} |
| (hh : Measurable h) {A : ℝ} (hA : 0≤A) (hhA : ∀ x,|h x|≤A) : |
| let b := gaussianField (n+1) (fun i (x : NormalizedSpin (n+1)) => x.val i) |
| let H := fun g x => h x+f.f (b g x) |
| (∫ g,gibbsReplicaMean μ (H g) 1 (fun x => b g (x 0)*f.d1 (b g (x 0))) |
| ∂Measure.pi (fun _ => gaussianReal 0 1))= |
| ∫ g,gibbsReplicaMean μ (H g) 1 (fun x => f.d2 (b g (x 0))+f.d1 (b g (x 0))^2)- |
| gibbsReplicaMean μ (H g) 2 (fun x => spinOverlap (x 0) (x 1)*f.d1 (b g (x 0))*f.d1 (b g (x 1))) |
| ∂Measure.pi (fun _ => gaussianReal 0 1) := by |
| dsimp only |
| let v := fun i (x : NormalizedSpin (n+1)) => x.val i |
| let w := fun i (x : Fin 1→NormalizedSpin (n+1)) => v i (x 0) |
| have hv : ∀ i,Measurable (v i) := fun i => by dsimp [v]; fun_prop |
| have hw : ∀ i,Measurable (w i) := fun i => by dsimp [w,v]; fun_prop |
| have hK (x : Fin 1→NormalizedSpin (n+1)) (y : NormalizedSpin (n+1)) : |
| (∑ i,w i x*v i y)=spinOverlap (x 0) y := by simp only [w,v,spinOverlap_sum] |
| have hb (g : Fin (n+1)→ℝ) (x : Fin 1→NormalizedSpin (n+1)) : |
| (∑ i,g i*w i x)=gaussianField (n+1) v g (x 0) := rfl |
| have he := nonlinear_pattern_direction_ibp μ (0:Fin 1) f hv hh hw hA |
| (by norm_num : (0:ℝ)≤1) (by norm_num : (0:ℝ)≤1) hhA |
| (fun i x => normalizedSpin_coord_bound x i) (fun i x => normalizedSpin_coord_bound (x 0) i) |
| dsimp only at he |
| simp_rw [hK,hb] at he |
| convert he using 1 |
| · congr 1; funext g; congr 1; funext x; ring |
| · congr 1; funext g |
| simp only [Nat.cast_one,one_mul,Fin.sum_univ_one,spinOverlap_self,Fin.succ_zero_eq_one,mul_one] |
| congr 1 |
| · congr 1; funext x; ring |
| · congr 1; funext x |
| rw [show spinOverlap (x 1) (x 0)=spinOverlap (x 0) (x 1) from real_inner_comm _ _] |
| ring |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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