| import Mathlib |
| import OAI.Probability.Perceptron.Cavity.CavityBulkCoefficients |
| import OAI.Probability.Perceptron.Cavity.CavitySphericalMoments |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped Topology |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma finiteGaussian_coordinate_projection {I J : Type*} [Fintype I] [Fintype J] |
| (e : J→I) (he : Function.Injective e) : |
| MeasurePreserving (fun x : EuclideanSpace ℝ I=>fun j=>x (e j)) |
| (stdGaussian (EuclideanSpace ℝ I)) (Measure.pi fun _ : J=>gaussianReal 0 1) := by |
| refine ⟨by fun_prop,?_⟩ |
| rw [←map_pi_eq_stdGaussian,Measure.map_map (by fun_prop) (PiLp.continuous_toLp 2 _).measurable] |
| change (Measure.pi fun _ : I=>gaussianReal 0 1).map (fun x j=>x (e j))=_ |
| rw [←Measure.infinitePi_eq_pi,Measure.map_infinitePi_infinitePi_of_inj he,Measure.infinitePi_eq_pi] |
|
|
| lemma finiteGaussian_curry_preserving (I J : Type*) [Fintype I] [Fintype J] : |
| MeasurePreserving (fun x : I×J→ℝ=>fun i j=>x (i,j)) |
| (Measure.pi fun _ : I×J=>gaussianReal 0 1) |
| (Measure.pi fun _ : I=>Measure.pi fun _ : J=>gaussianReal 0 1) := by |
| refine ⟨by fun_prop,?_⟩ |
| have h:=Measure.infinitePi_map_curry (μ:=fun (_ : I) (_ : J)=>gaussianReal 0 1) |
| simp only [Measure.infinitePi_eq_pi] at h |
| convert h using 1 |
| rfl |
|
|
| def cavityBulkNoiseSplit (n d m M : ℕ) |
| (y : EuclideanSpace ℝ (CavityBulkNoiseIndex n d m M)) : |
| (Fin M→Spin (n+1))×(Fin d→Fin (m+1)→ℝ) := |
| (fun t=>WithLp.toLp 2 (fun i=>y (Sum.inl i,Sum.inl t)), |
| fun t i=>y (Sum.inr t,Sum.inr i)) |
|
|
| lemma cavityBulkNoiseSplit_preserving (n d m M : ℕ) : |
| MeasurePreserving (cavityBulkNoiseSplit n d m M) |
| (stdGaussian (EuclideanSpace ℝ (CavityBulkNoiseIndex n d m M))) |
| ((Measure.pi fun _ : Fin M=>stdGaussian (Spin (n+1))).prod |
| (finitePatternRowsLaw (m+1) d)) := by |
| let e : (Fin M×Fin (n+1))⊕(Fin d×Fin (m+1))→CavityBulkNoiseIndex n d m M := |
| Sum.elim (fun p=>(Sum.inl p.2,Sum.inl p.1)) (fun p=>(Sum.inr p.1,Sum.inr p.2)) |
| have he : Function.Injective e := by |
| intro a b h |
| cases a <;> cases b <;> simp only [e,Sum.elim_inl,Sum.elim_inr,Prod.mk.injEq,Sum.inl.injEq, |
| Sum.inr.injEq,Sum.inl_ne_inr,Sum.inr_ne_inl,false_and] at h |
| · exact congrArg Sum.inl (Prod.ext h.2 h.1) |
| · exact congrArg Sum.inr (Prod.ext h.1 h.2) |
| have hp:=(measurePreserving_sumPiEquivProdPi |
| (fun _ : (Fin M×Fin (n+1))⊕(Fin d×Fin (m+1))=>gaussianReal 0 1)).comp |
| (finiteGaussian_coordinate_projection e he) |
| have ht : MeasurePreserving (WithLp.toLp 2 : (Fin (n+1)→ℝ)→Spin (n+1)) |
| (Measure.pi fun _=>gaussianReal 0 1) (stdGaussian (Spin (n+1))) := |
| ⟨(PiLp.continuous_toLp 2 _).measurable,map_pi_eq_stdGaussian⟩ |
| have hleft:=(measurePreserving_pi (fun _ : Fin M=>Measure.pi fun _ : Fin (n+1)=>gaussianReal 0 1) |
| (fun _=>stdGaussian (Spin (n+1))) (fun _=>ht)).comp (finiteGaussian_curry_preserving (Fin M) (Fin (n+1))) |
| convert (hleft.prod (finiteGaussian_curry_preserving (Fin d) (Fin (m+1)))).comp hp using 1 |
| · rfl |
| · rfl |
|
|
| lemma cavityBulkNoiseSplit_scalar (n d m M : ℕ) (f : Jet3) (a : BulkDisorder (m+1) M) |
| (x : NormalizedSpin (m+1)) (y : EuclideanSpace ℝ (CavityBulkNoiseIndex n d m M)) (t : Fin d) : |
| inner ℝ (cavityBulkFeature n d m M f a x (Sum.inr t)) y= |
| ∑ i,(cavityBulkNoiseSplit n d m M y).2 t i*x.val i := by |
| classical |
| simp only [EuclideanSpace.inner_eq_star_dotProduct,dotProduct,cavityBulkFeature, |
| star_trivial,Fintype.sum_prod_type] |
| simp only [mul_ite,mul_zero,cavityBulkBase,Fintype.sum_sum_type,cavityBulkNoiseSplit] |
| simp only [Sum.inl_ne_inr,Sum.inr.injEq,ite_false,ite_self,Finset.sum_const_zero,zero_add] |
| rw [Finset.sum_comm] |
| simp |
|
|
| lemma cavityBulkNoiseSplit_vector (n d m M : ℕ) (f : Jet3) (a : BulkDisorder (m+1) M) |
| (x : NormalizedSpin (m+1)) (y : EuclideanSpace ℝ (CavityBulkNoiseIndex n d m M)) : |
| gaussianRows (fun i=>cavityBulkFeature n d m M f a x (Sum.inl i)) y= |
| cavityVectorField M (n+1) (fun t=>f.d1 (∑ i,a.1 t i*x.val i)) |
| (Real.sqrt (m+1:ℕ))⁻¹ (cavityBulkNoiseSplit n d m M y).1 := by |
| classical |
| ext i |
| simp only [gaussianRows_apply,EuclideanSpace.inner_eq_star_dotProduct,dotProduct,cavityBulkFeature, |
| star_trivial,Fintype.sum_prod_type] |
| simp only [mul_ite,mul_zero,cavityBulkBase,Fintype.sum_sum_type, |
| cavityBulkNoiseSplit,cavityVectorField,PiLp.smul_apply,smul_eq_mul] |
| simp only [EuclideanSpace.inner_eq_star_dotProduct,dotProduct,star_trivial] |
| simp only [Sum.inl.injEq,Sum.inr_ne_inl,ite_false,ite_self,Finset.sum_const_zero,add_zero] |
| rw [Finset.sum_comm] |
| simp only [Fintype.sum_ite_eq'] |
| try simp only [cavityVectorField,PiLp.smul_apply,smul_eq_mul,PiLp.sum_apply,PiLp.toLp_apply] |
| simp_rw [mul_comm (y.ofLp _) _] |
| simp only [WithLp.ofLp_sum,PiLp.smul_apply,Finset.sum_apply,smul_eq_mul] |
| simp only [mul_assoc,Finset.mul_sum] |
| end SphericalPerceptronFreeEnergy |
| |
| end |
| |
| end OAI |
| |
