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import OAI.Probability.Perceptron.Variational.CovarianceMatrix
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped Topology BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
variable {K L : Type*} [TopologicalSpace K] [TopologicalSpace L]
def compactMapArray (e : K→L) (Q : CompactArray K) : CompactArray L := fun i j => e (Q i j)
def compactMapBlock (e : K→L) (n : ℕ) (Q : CompactBlock K n) : CompactBlock L n := fun i j => e (Q i j)
lemma compactMapArray_continuous {e : K→L} (he : Continuous e) : Continuous (compactMapArray e) := by
unfold compactMapArray
fun_prop
lemma compactMapBlock_continuous {e : K→L} (he : Continuous e) (n : ℕ) :
Continuous (compactMapBlock e n) := by
unfold compactMapBlock
fun_prop
def compactDiagonalMapArray (e d : K→L) (Q : CompactArray K) : CompactArray L :=
fun i j => if i=j then d (Q i j) else e (Q i j)
def compactDiagonalMapBlock (e d : K→L) (n : ℕ) (Q : CompactBlock K n) : CompactBlock L n :=
fun i j => if i=j then d (Q i j) else e (Q i j)
lemma compactDiagonalMapArray_continuous {e d : K→L} (he : Continuous e) (hd : Continuous d) :
Continuous (compactDiagonalMapArray e d) := by
apply continuous_pi; intro i; apply continuous_pi; intro j
unfold compactDiagonalMapArray
split_ifs <;> fun_prop
lemma compactDiagonalMapBlock_continuous {e d : K→L} (he : Continuous e) (hd : Continuous d) (n : ℕ) :
Continuous (compactDiagonalMapBlock e d n) := by
apply continuous_pi; intro i; apply continuous_pi; intro j
unfold compactDiagonalMapBlock
split_ifs <;> fun_prop
omit [TopologicalSpace K] [TopologicalSpace L] in
lemma compactDiagonalMapBlock_compactBlock (e d : K→L) (n : ℕ) (Q : CompactArray K) :
compactBlock n (compactDiagonalMapArray e d Q)=compactDiagonalMapBlock e d n (compactBlock n Q) := by
funext i j
simp only [compactBlock,compactDiagonalMapArray,compactDiagonalMapBlock,Fin.val_inj]
variable [MeasurableSpace K] [MeasurableSpace L] [BorelSpace K] [BorelSpace L]
[SecondCountableTopology K] [SecondCountableTopology L]
def compactMapLaw (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K)) :
ProbabilityMeasure (CompactArray L) :=
⟨(μ : Measure (CompactArray K)).map (compactMapArray e),
(Measure.isProbabilityMeasure_map_iff (compactMapArray_continuous he).measurable.aemeasurable).2
inferInstance⟩
lemma compactMapLaw_integral (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K))
(F : CompactArray L→ℝ) (hF : Measurable F) :
(∫ Q, F Q ∂(compactMapLaw e he μ : Measure (CompactArray L)))=
∫ Q, F (compactMapArray e Q) ∂(μ : Measure (CompactArray K)) :=
integral_map (compactMapArray_continuous he).measurable.aemeasurable hF.aestronglyMeasurable
lemma compactMapLaw_gg (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K))
(hGG : ∀ (n : ℕ) (i : Fin n) (f : CompactBlock K n →ᵇ ℝ) (g : K →ᵇ ℝ),
compactGGDefect μ n i f g=0) (n : ℕ) (i : Fin n)
(f : CompactBlock L n →ᵇ ℝ) (g : L →ᵇ ℝ) :
compactGGDefect (compactMapLaw e he μ) n i f g=0 := by
let f' := f.compContinuous ⟨compactMapBlock e n,compactMapBlock_continuous he n⟩
let g' := g.compContinuous ⟨e,he⟩
have hmul (a b : ℕ) :
(∫ Q, f (compactBlock n Q)*g (Q a b) ∂(compactMapLaw e he μ : Measure (CompactArray L)))=
∫ Q, f' (compactBlock n Q)*g' (Q a b) ∂(μ : Measure (CompactArray K)) := by
exact compactMapLaw_integral e he μ _
((f.measurable.comp (compactBlock_continuous n).measurable).mul (g.measurable.comp (by fun_prop)))
have hf : (∫ Q, f (compactBlock n Q) ∂(compactMapLaw e he μ : Measure (CompactArray L)))=
∫ Q, f' (compactBlock n Q) ∂(μ : Measure (CompactArray K)) := by
exact compactMapLaw_integral e he μ _
(f.measurable.comp (compactBlock_continuous n).measurable)
have hg : (∫ Q, g (Q 0 1) ∂(compactMapLaw e he μ : Measure (CompactArray L)))=
∫ Q, g' (Q 0 1) ∂(μ : Measure (CompactArray K)) := by
exact compactMapLaw_integral e he μ _ (by fun_prop)
unfold compactGGDefect
simp_rw [hmul,hf,hg]
exact hGG n i f' g'
lemma compactMapLaw_pair (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K))
(a b : ℕ) :
(compactMapLaw e he μ : Measure (CompactArray L)).map (fun Q => Q a b)=
((μ : Measure (CompactArray K)).map (fun Q => Q a b)).map e := by
change ((μ : Measure (CompactArray K)).map (compactMapArray e)).map (fun Q => Q a b)=_
rw [Measure.map_map (by fun_prop) (compactMapArray_continuous he).measurable,
Measure.map_map he.measurable (by fun_prop)]
rfl
def compactDiagonalMapLaw (e d : K→L) (he : Continuous e) (hd : Continuous d)
(μ : ProbabilityMeasure (CompactArray K)) : ProbabilityMeasure (CompactArray L) :=
⟨(μ : Measure (CompactArray K)).map (compactDiagonalMapArray e d),
(Measure.isProbabilityMeasure_map_iff
(compactDiagonalMapArray_continuous he hd).measurable.aemeasurable).2 inferInstance⟩
lemma compactDiagonalMapLaw_integral (e d : K→L) (he : Continuous e) (hd : Continuous d)
(μ : ProbabilityMeasure (CompactArray K)) (F : CompactArray L→ℝ) (hF : Measurable F) :
(∫ Q, F Q ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))=
∫ Q, F (compactDiagonalMapArray e d Q) ∂(μ : Measure (CompactArray K)) :=
integral_map (compactDiagonalMapArray_continuous he hd).measurable.aemeasurable hF.aestronglyMeasurable
lemma compactDiagonalMapLaw_gg (e d : K→L) (he : Continuous e) (hd : Continuous d)
(μ : ProbabilityMeasure (CompactArray K))
(hGG : ∀ (n : ℕ) (i : Fin n) (f : CompactBlock K n →ᵇ ℝ) (g : K →ᵇ ℝ),
compactGGDefect μ n i f g=0) (n : ℕ) (i : Fin n)
(f : CompactBlock L n →ᵇ ℝ) (g : L →ᵇ ℝ) :
compactGGDefect (compactDiagonalMapLaw e d he hd μ) n i f g=0 := by
let f' := f.compContinuous ⟨compactDiagonalMapBlock e d n,compactDiagonalMapBlock_continuous he hd n⟩
let g' := g.compContinuous ⟨e,he⟩
have hmul (a b : ℕ) (hab : a≠b) :
(∫ Q, f (compactBlock n Q)*g (Q a b) ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))=
∫ Q, f' (compactBlock n Q)*g' (Q a b) ∂(μ : Measure (CompactArray K)) := by
rw [compactDiagonalMapLaw_integral e d he hd μ (fun Q => f (compactBlock n Q)*g (Q a b))
((f.measurable.comp (compactBlock_continuous n).measurable).mul (g.measurable.comp (by fun_prop)))]
simp only [compactDiagonalMapBlock_compactBlock,compactDiagonalMapArray,ite_eq_right hab]
rfl
have hf : (∫ Q, f (compactBlock n Q) ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))=
∫ Q, f' (compactBlock n Q) ∂(μ : Measure (CompactArray K)) := by
rw [compactDiagonalMapLaw_integral e d he hd μ (fun Q => f (compactBlock n Q))
(f.measurable.comp (compactBlock_continuous n).measurable)]
simp only [compactDiagonalMapBlock_compactBlock]
rfl
have hg : (∫ Q, g (Q 0 1) ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))=
∫ Q, g' (Q 0 1) ∂(μ : Measure (CompactArray K)) := by
rw [compactDiagonalMapLaw_integral e d he hd μ (fun Q => g (Q 0 1)) (by fun_prop)]
rfl
unfold compactGGDefect
rw [hmul i.val n (Nat.ne_of_lt i.isLt),hf,hg]
have hs : (∑ j ∈ Finset.univ.erase i, ∫ Q, f (compactBlock n Q)*g (Q i j)
∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))=
∑ j ∈ Finset.univ.erase i, ∫ Q, f' (compactBlock n Q)*g' (Q i j)
∂(μ : Measure (CompactArray K)) := by
apply Finset.sum_congr rfl
intro j hj
apply hmul
intro hij
exact (Finset.mem_erase.mp hj).1 (Fin.ext hij.symm)
rw [hs]
exact hGG n i f' g'
end SphericalPerceptronFreeEnergy
end
end OAI
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