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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
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