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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 | import Mathlib
import OAI.Probability.Perceptron.Interpolation.GibbsOverlapArray
namespace OAI
noncomputable section
namespace SphericalPerceptronFreeEnergy
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
def compactJointMonomial (p d : ℕ) : CompactJointOverlap →ᵇ ℝ :=
BoundedContinuousFunction.mkOfCompact ⟨fun x : CompactJointOverlap => (x.2:ℝ)^d*(x.1:ℝ)^p,by fun_prop⟩
@[simp] lemma compactJointMonomial_apply (p d : ℕ) (x : CompactJointOverlap) :
compactJointMonomial p d x = (x.2:ℝ)^d*(x.1:ℝ)^p := rfl
lemma compactJointMonomial_zero : compactJointMonomial 0 0 = 1 := by ext x; simp
lemma compactJointMonomial_add (p d p' d' : ℕ) :
compactJointMonomial (p+p') (d+d') = compactJointMonomial p d * compactJointMonomial p' d' := by
ext x
simp only [compactJointMonomial_apply,BoundedContinuousFunction.mul_apply,pow_add]
ring
lemma compactJointMonomial_separates (x y : CompactJointOverlap) (h : x≠y) :
∃ p d, compactJointMonomial p d x ≠ compactJointMonomial p d y := by
by_contra hn
push Not at hn
apply h
apply Prod.ext <;> apply Subtype.ext
· simpa using hn 1 0
· simpa using hn 0 1
lemma compactJointMonomial_abs_le (p d : ℕ) (x : CompactJointOverlap) :
|compactJointMonomial p d x|≤1 := by
rw [compactJointMonomial_apply,abs_mul,abs_pow,abs_pow,abs_of_nonneg x.2.property.1]
exact (mul_le_of_le_one_left (by positivity)
(pow_le_one₀ x.2.property.1 x.2.property.2)).trans
(pow_le_one₀ (abs_nonneg _) (abs_le.mpr x.1.property))
lemma compactJoint_monomials_determine (P Q : Measure CompactJointOverlap)
[IsFiniteMeasure P] [IsFiniteMeasure Q]
(h : ∀ p d, (∫ x, compactJointMonomial p d x ∂P) = ∫ x, compactJointMonomial p d x ∂Q) : P=Q := by
apply measure_eq_of_additive_test_family (fun a : ℕ×ℕ => compactJointMonomial a.1 a.2)
compactJointMonomial_zero (fun a b => compactJointMonomial_add _ _ _ _)
· intro x y hxy
obtain ⟨p,d,hpd⟩ := compactJointMonomial_separates x y hxy
exact ⟨(p,d),hpd⟩
· exact fun a => h a.1 a.2
lemma sourceJointOverlap_symm {N k : ℕ} (x y : NormalizedSpin N×IndexedLeaf k) :
sourceJointOverlap x y = sourceJointOverlap y x := by
apply Prod.ext <;> apply Subtype.ext
· exact real_inner_comm _ _
· simp only [sourceJointOverlap,indexedCommonDepth_comm k y.2 x.2]
lemma sourceGibbsArray_eq (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ)
(h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (z : Fin k → ℝ) (t : ℝ≥0) :
sourceGibbsArrayLaw n k f p d h u z t =
gibbsOverlapArrayLaw (sourceFullSpinLeafKernel n k)
((sourceBaseDataLaw n k z t).prod countableGaussianLaw)
(sourceCouplingHamiltonian n k f p d h u)
(sourceCouplingHamiltonian_measurable n k f p d h u)
sourceJointOverlap (sourceJointOverlap_measurable (n+1) k) := rfl
lemma source_compact_gg_defect (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ)
(h : Fin (k+1) → ℝ) (hh0 : ∀ l, 0≤h l) (hh : Monotone h)
(u : Fin (n+1) → ℝ) (j : Fin (n+1)) (z : Fin k → ℝ) (t : ℝ≥0)
(r : ℕ) (i : Fin r) (G : CompactBlock CompactJointOverlap r →ᵇ ℝ) :
compactGGDefect (sourceGibbsArrayLaw n k f p d h u z t) r i G (compactJointMonomial (p j) (d j)) =
-kernelGGDefect (sourceFullSpinLeafKernel n k)
((sourceBaseDataLaw n k z t).prod countableGaussianLaw)
(sourceCouplingHamiltonian n k f p d h u) (sourceJointMonomial p d j)
(((k:ℝ)/(k+1:ℕ))^(d j)) r i
(fun x => G (fun a b => sourceJointOverlap (x a) (x b))) := by
rw [sourceGibbsArray_eq]
have he := sourceCoupling_all_exp_ae n k f p d h hh0 hh u 0 z t
have hexp : ∀ᵐ a ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw,
Integrable (fun x => Real.exp (sourceCouplingHamiltonian n k f p d h u a x))
(sourceFullSpinLeafKernel n k a) := by
filter_upwards [he] with a ha
simpa only [sourceFullSpinLeafKernel,Kernel.comap_apply,zero_mul,add_zero] using ha 0
apply compactGGDefect_gibbsOverlapArray _ _ _ _ _ _ hexp r i G (compactJointMonomial (p j) (d j))
· exact sourceJointMonomial_diagonal p d j
· intro x y
rw [sourceJointOverlap_symm x y]
end SphericalPerceptronFreeEnergy
end
end OAI
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