| import Mathlib |
| import OAI.Probability.Perceptron.Variational.CompactGGDensity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def compactScalarMonomial (p : ℕ) : CompactOverlap →ᵇ ℝ := |
| BoundedContinuousFunction.mkOfCompact ⟨fun x : CompactOverlap => (x:ℝ)^p,by fun_prop⟩ |
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| @[simp] lemma compactScalarMonomial_apply (p : ℕ) (x : CompactOverlap) : |
| compactScalarMonomial p x = (x:ℝ)^p := rfl |
|
|
| lemma compactScalarMonomial_add (p q : ℕ) : |
| compactScalarMonomial (p+q)=compactScalarMonomial p*compactScalarMonomial q := by |
| ext x |
| simp only [compactScalarMonomial_apply,BoundedContinuousFunction.mul_apply,pow_add] |
|
|
| lemma compactScalar_monomials_total (L : (CompactOverlap →ᵇ ℝ) →L[ℝ] ℝ) |
| (hL : ∀ p, L (compactScalarMonomial p)=0) : L=0 := by |
| let e := ContinuousMap.linearIsometryBoundedOfCompact CompactOverlap ℝ ℝ |
| let L' := L.comp e.toContinuousLinearEquiv.toContinuousLinearMap |
| let m (a : ℕ) : C(CompactOverlap,ℝ) := (compactScalarMonomial a).toContinuousMap |
| have hm0 : m 0=1 := by ext x; simp [m,compactScalarMonomial_apply] |
| have hmadd (a b : ℕ) : m (a+b)=m a*m b := by |
| ext x |
| exact congrArg (fun f : CompactOverlap →ᵇ ℝ => f x) (compactScalarMonomial_add _ _) |
| let M : Submonoid C(CompactOverlap,ℝ) := |
| { carrier := Set.range m |
| one_mem' := ⟨0,hm0⟩ |
| mul_mem' := by rintro _ _ ⟨a,rfl⟩ ⟨b,rfl⟩; exact ⟨a+b,hmadd a b⟩ } |
| let A := Algebra.adjoin ℝ (M : Set C(CompactOverlap,ℝ)) |
| have hsep : A.SeparatesPoints := by |
| intro x y hxy |
| have hpd : compactScalarMonomial 1 x ≠ compactScalarMonomial 1 y := by |
| simpa only [compactScalarMonomial_apply,pow_one] using (Subtype.val_injective.ne hxy) |
| have hmem : m 1 ∈ A := |
| Algebra.subset_adjoin (show m 1 ∈ M from ⟨1,rfl⟩) |
| exact ⟨m 1,⟨m 1,hmem,rfl⟩,hpd⟩ |
| have hA : (A : Set C(CompactOverlap,ℝ)) ⊆ (L'.ker : Set C(CompactOverlap,ℝ)) := by |
| have hspan : Submodule.span ℝ (M : Set C(CompactOverlap,ℝ)) ≤ L'.ker := by |
| apply Submodule.span_le.mpr |
| rintro _ ⟨p,rfl⟩ |
| change L (e ((compactScalarMonomial p).toContinuousMap)) = 0 |
| have he : e ((compactScalarMonomial p).toContinuousMap) = compactScalarMonomial p := by |
| ext x; rfl |
| rw [he] |
| exact hL p |
| intro f hf |
| apply hspan |
| change f ∈ A.toSubmodule at hf |
| simpa only [A,Algebra.adjoin_eq_span,Submonoid.closure_eq] using hf |
| have hclosed := L'.isClosed_ker |
| have hc := closure_minimal hA hclosed |
| have htop := ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints A hsep |
| have hall (f : C(CompactOverlap,ℝ)) : L' f=0 := by |
| apply hc |
| change f ∈ A.topologicalClosure |
| rw [htop] |
| trivial |
| ext g |
| have h := hall (e.symm g) |
| simpa [L'] using h |
|
|
| lemma compactGG_of_scalar_monomials (μ : ProbabilityMeasure (CompactArray CompactOverlap)) |
| (n : ℕ) (i : Fin n) (f : CompactBlock CompactOverlap n →ᵇ ℝ) |
| (h : ∀ p, compactGGDefect μ n i f (compactScalarMonomial p)=0) |
| (g : CompactOverlap →ᵇ ℝ) : compactGGDefect μ n i f g=0 := by |
| have hz := compactScalar_monomials_total (compactGGTestCLM μ n i f) |
| (fun p => (compactGGTestCLM_apply μ n i f _).trans (h p)) |
| rw [← compactGGTestCLM_apply,hz] |
| rfl |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
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|
