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c4cdbbd | 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 99 100 101 | import OAI.Probability.InvariantIsing.Fields.FieldFiniteLocal
import OAI.Probability.InvariantIsing.Fields.FieldFiniteMeasurable
namespace OAI
/-! Differentiation of the actual finite-height Gaussian transform and
its tilted observables. The local envelope is derived from the family. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
namespace FieldFiniteFamily
variable {n : ℕ} {I : Set (Fin n → ℝ)} (F : FieldFiniteFamily n I)
lemma affine_average_hasFDerivAt (hI : IsOpen I)
(A : FieldCovariate n → ℝ → ℝ)
(DA : FieldCovariate n → ℝ → FieldCovariate n →L[ℝ] ℝ)
(a : ℝ) (v : Fin n → ℝ) (ζ : ℝ) {V R : ℝ} (hR : 0 ≤ R)
(hlo : ∀ t ∈ I, 0 < fieldFiniteVariance a v t)
(hhi : ∀ t ∈ I, fieldFiniteVariance a v t ≤ V)
(hc : ∀ t ∈ I, ∀ i, |fieldFiniteSlope a v t i| ≤ R)
{CA CE : ℝ} (hCA : 0 ≤ CA) (hCE : 0 ≤ CE)
(hA : ∀ q, Measurable (A q))
(hDA : ∀ q, AEStronglyMeasurable (DA q) (gaussianReal 0 1))
(hAb : ∀ q, q.1 ∈ I → ∀ u, |A q u| ≤ CA * (1 + |u|))
(hEb : ∀ q, q.1 ∈ I → ∀ u, ‖DA q u‖ ≤ CE * (1 + |u|) ^ 2)
(dA : ∀ q, q.1 ∈ I → ∀ u, HasFDerivAt (fun r => A r u) (DA q u) q)
{p : FieldCovariate n} (hp : p.1 ∈ I) :
HasFDerivAt (fun q => ∫ u, A q u ∂F.affineLaw a v ζ q)
((∫ u, DA p u + (ζ * A p u) • F.affineDifferential a v p u ∂F.affineLaw a v ζ p) -
(∫ u, A p u ∂F.affineLaw a v ζ p) •
(∫ u, ζ • F.affineDifferential a v p u ∂F.affineLaw a v ζ p)) p := by
obtain ⟨S, C, L, CD, hS, hpS, hSI, hCD, hExp, hD⟩ :=
F.local_envelope hI a v ζ hR hhi hc hp
apply field_finite_gaussianPair_hasFDerivAt (F.affineShift a v) A
(F.affineDifferential a v) DA hS hpS ζ C L CA CD CE hCA hCD hCE
(fun q => F.mU.comp (by dsimp only [shiftPoint]; fun_prop)) hA
(F.measurable_affineDifferential a v p).aestronglyMeasurable (hDA p) hExp
(fun q hq => hAb q (hSI q hq)) hD (fun q hq => hEb q (hSI q hq))
· intro q hq u
exact field_finite_affine_shift_hasFDerivAt F.derivative a v u
(hSI q hq) (hlo q.1 (hSI q hq))
· exact fun q hq => dA q (hSI q hq)
def value (a : ℝ) (v : Fin n → ℝ) (ζ : ℝ) (p : FieldCovariate n) : ℝ :=
gaussianTransform (fieldFiniteVariance a v p.1) ζ (fun y => F.U (p.1, y)) p.2
lemma value_hasFDerivAt (hI : IsOpen I) (a : ℝ) (v : Fin n → ℝ) (ζ : ℝ)
{V R : ℝ} (hR : 0 ≤ R)
(hlo : ∀ t ∈ I, 0 < fieldFiniteVariance a v t)
(hhi : ∀ t ∈ I, fieldFiniteVariance a v t ≤ V)
(hc : ∀ t ∈ I, ∀ i, |fieldFiniteSlope a v t i| ≤ R)
{p : FieldCovariate n} (hp : p.1 ∈ I) :
HasFDerivAt (F.value a v ζ)
(∫ u, F.affineDifferential a v p u ∂F.affineLaw a v ζ p) p := by
obtain ⟨S, C, L, CD, hS, _hpS, hSI, hCD, hExp, hD⟩ :=
F.local_envelope hI a v ζ hR hhi hc hp
let B : ℝ → ℝ := fun u => CD * (1 + |u|)
let E : ℝ → ℝ := fun u => Real.exp (C + L * |u|)
have hBg : HasLinearGrowth B := by
refine ⟨CD, CD, hCD, hCD, fun u => ?_⟩
dsimp only [B]
rw [abs_of_nonneg (by positivity), Real.norm_eq_abs]
exact le_of_eq (by ring)
have hBi : Integrable B (gaussianReal 0 1) :=
field_gaussian_linear_integrable 1 0 (by fun_prop) hBg
have hEBi : Integrable (fun u => E u * B u) (gaussianReal 0 1) := by
apply ((field_gaussian_quadratic_envelope_integrable C L).const_mul CD).mono' (by fun_prop)
apply ae_of_all
intro u
change |Real.exp (C + L * |u|) * (CD * (1 + |u|))| ≤
CD * (Real.exp (C + L * |u|) * (1 + |u|) ^ 2)
rw [abs_of_nonneg (by positivity)]
calc
_ = CD * Real.exp (C + L * |u|) * (1 + |u|) := by ring
_ ≤ CD * Real.exp (C + L * |u|) * (1 + |u|) ^ 2 :=
mul_le_mul_of_nonneg_left (field_mark_one_add_le_sq u) (by positivity)
_ = _ := by ring
have hm : Measurable (F.affineShift a v p) := F.mU.comp (by dsimp only [shiftPoint]; fun_prop)
have hg : HasLinearGrowth (F.affineShift a v p) :=
(F.growth p.1 hp).add_left p.2 |>.scale_argument (Real.sqrt (fieldFiniteVariance a v p.1))
exact field_finite_average_hasFDerivAt_of_envelope (gaussianReal 0 1)
(F.affineShift a v) (F.affineDifferential a v) hS
(fun q => F.mU.comp (by dsimp only [shiftPoint]; fun_prop))
(F.measurable_affineDifferential a v p).aestronglyMeasurable
(field_gaussian_linear_integrable 1 0 hm hg) ζ
(integrable_exp_of_linearGrowth _ (gaussianReal_exponentialNormMoments 0 1) hm hg ζ)
B E hBi hEBi hD hExp (fun q hq u =>
field_finite_affine_shift_hasFDerivAt F.derivative a v u
(hSI q hq) (hlo q.1 (hSI q hq)))
end FieldFiniteFamily
end InvariantIsing
end
end OAI
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