| import Mathlib |
| import OAI.Probability.Perceptron.Sphere.SphereIsometryMap |
| import OAI.Probability.Perceptron.Variational.TensorFeatures |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology BigOperators |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def unitSphereAmbientLaw (N : ℕ) : Measure (Spin N) := |
| (unitSphereLaw N).map (fun x : NormalizedSpin N => x.val) |
| |
| instance unitSphereAmbientLaw_probability (n : ℕ) : IsProbabilityMeasure (unitSphereAmbientLaw (n+1)) := by |
| unfold unitSphereAmbientLaw |
| apply (Measure.isProbabilityMeasure_map_iff measurable_subtype_coe.aemeasurable).2 |
| have : NeZero ((volume : Measure (Spin (n+1))).toSphere) := |
| ⟨Measure.toSphere_ne_zero volume⟩ |
| unfold unitSphereLaw |
| infer_instance |
| |
| lemma unitSphereAmbientLaw_norm (N : ℕ) : ∀ᵐ x ∂unitSphereAmbientLaw N, ‖x‖=1 := by |
| apply (ae_map_iff measurable_subtype_coe.aemeasurable (measurableSet_eq_fun continuous_norm.measurable measurable_const)).mpr |
| exact ae_of_all _ fun x => by simpa only [Metric.mem_sphere,dist_zero_right] using x.property |
| |
| lemma unitSphereAmbientLaw_preserving {N : ℕ} (e : Spin N≃ₗᵢ[ℝ]Spin N) : |
| MeasurePreserving e (unitSphereAmbientLaw N) (unitSphereAmbientLaw N) := by |
| refine ⟨e.continuous.measurable,?_⟩ |
| rw [unitSphereAmbientLaw,Measure.map_map e.continuous.measurable measurable_subtype_coe] |
| have he : e ∘ (fun x : NormalizedSpin N => x.val) = |
| (fun x : NormalizedSpin N => x.val) ∘ sphereIsometryMap e := rfl |
| rw [he,← Measure.map_map measurable_subtype_coe (sphereIsometryMap_measurable e), |
| (unitSphereLaw_isometry_preserving e).map_eq] |
| |
| lemma unitSphere_tangent_integral (n : ℕ) (y : Spin (n+1)) {F : Spin (n+1)→ℝ} |
| (hF : ContDiff ℝ 1 F) : |
| (∫ x : NormalizedSpin (n+1), fderiv ℝ F x.val (y-inner ℝ y x.val • x.val) ∂unitSphereLaw (n+1)) = |
| (n:ℝ)*(∫ x : NormalizedSpin (n+1), inner ℝ y x.val*F x.val ∂unitSphereLaw (n+1)) := by |
| have he := invariant_tangent_integral (unitSphereAmbientLaw (n+1)) |
| ((unitSphereAmbientLaw_norm (n+1)).mono fun _ h => h.le) |
| (fun e => unitSphereAmbientLaw_preserving e) (Fin (n+1)) |
| (EuclideanSpace.basisFun (Fin (n+1)) ℝ) y hF |
| have hleft : AEStronglyMeasurable (fun x : Spin (n+1) => fderiv ℝ F x ((‖x‖^2) • y-inner ℝ y x • x)) |
| (unitSphereAmbientLaw (n+1)) := |
| ((hF.continuous_fderiv (by norm_num)).clm_apply (by fun_prop)).aestronglyMeasurable |
| have hright : AEStronglyMeasurable (fun x : Spin (n+1) => ((Fintype.card (Fin (n+1)):ℝ)-1)*inner ℝ y x*F x) |
| (unitSphereAmbientLaw (n+1)) := (by fun_prop : Continuous _).aestronglyMeasurable |
| rw [unitSphereAmbientLaw,integral_map measurable_subtype_coe.aemeasurable hleft, |
| integral_map measurable_subtype_coe.aemeasurable hright] at he |
| have hn (x : NormalizedSpin (n+1)) : ‖x.val‖=1 := by |
| simpa only [Metric.mem_sphere,dist_zero_right] using x.property |
| simpa only [hn,one_pow,one_smul,Fintype.card_fin,Nat.cast_add,Nat.cast_one, |
| add_sub_cancel_right,mul_assoc,integral_const_mul] using he |
| |
| lemma unitSphere_gibbs_tangent_integral (n : ℕ) (y : NormalizedSpin (n+1)) |
| {H : Spin (n+1)→ℝ} (hH : ContDiff ℝ 1 H) {G : ℝ→ℝ} (hG : ContDiff ℝ 1 G) : |
| (n:ℝ)*(∫ x : NormalizedSpin (n+1), spinOverlap y x*(G (spinOverlap y x)*Real.exp (H x.val)) |
| ∂unitSphereLaw (n+1)) = |
| ∫ x : NormalizedSpin (n+1), |
| (deriv G (spinOverlap y x)*(1-spinOverlap y x^2)+G (spinOverlap y x)* |
| fderiv ℝ H x.val (y.val-spinOverlap y x • x.val))*Real.exp (H x.val) ∂unitSphereLaw (n+1) := by |
| let F : Spin (n+1)→ℝ := fun x => G (inner ℝ y.val x)*Real.exp (H x) |
| have hF : ContDiff ℝ 1 F := (hG.comp (innerSL ℝ y.val).contDiff).mul hH.exp |
| change (n:ℝ)*(∫ x : NormalizedSpin (n+1), inner ℝ y.val x.val*F x.val ∂unitSphereLaw (n+1)) = _ |
| rw [← unitSphere_tangent_integral n y.val hF] |
| apply integral_congr_ae |
| exact ae_of_all _ fun x => by |
| have hd := (((hG.differentiable (by norm_num) (inner ℝ y.val x.val)).hasDerivAt.comp_hasFDerivAt x.val |
| (innerSL ℝ y.val).hasFDerivAt).mul (hH.differentiable (by norm_num) x.val).hasFDerivAt.exp).fderiv |
| change fderiv ℝ F x.val (y.val-spinOverlap y x • x.val)=_ |
| have he : fderiv ℝ F x.val = G (spinOverlap y x) • (Real.exp (H x.val) • fderiv ℝ H x.val)+ |
| Real.exp (H x.val) • (deriv G (spinOverlap y x) • innerSL ℝ y.val) := by |
| simpa only [F,spinOverlap,Function.comp_def,Pi.mul_def] using hd |
| rw [he] |
| have hy : ‖y.val‖=1 := by simpa only [Metric.mem_sphere,dist_zero_right] using y.property |
| simp only [add_apply,smul_apply,smul_eq_mul, |
| coe_innerSL_apply,inner_sub_right,inner_smul_right,real_inner_self_eq_norm_sq,hy,one_pow,spinOverlap] |
| ring |
| |
| end SphericalPerceptronFreeEnergy |
| end |
| |
| end OAI |
| |
