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135052a | 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.SKBarriers.Hierarchy.HierarchyDensity
import OAI.Probability.SKBarriers.Scalar.GrowthOperations
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ExponentialAverage
variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
omit [NormedSpace ℝ E] in
theorem HasExpGrowth.gaussian_integral {f : E × ℝ → F} (hf : HasExpGrowth f) :
HasExpGrowth (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) := by
obtain ⟨C,M,hC,hM,hf⟩ := hf
let B := ∫ y : ℝ, Real.exp (M*|y|) ∂gaussianReal 0 1
have hB : 0 ≤ B := integral_nonneg (fun y => (Real.exp_pos _).le)
refine ⟨C*B,M,mul_nonneg hC hB,hM,?_⟩
intro x
have hb : ∀ y, ‖f (x,y)‖ ≤ C*Real.exp (M*‖x‖)*Real.exp (M*|y|) := by
intro y
apply (hf (x,y)).trans
rw [mul_assoc,← Real.exp_add,← mul_add]
apply mul_le_mul_of_nonneg_left _ hC
apply Real.exp_le_exp.2
apply mul_le_mul_of_nonneg_left _ hM
rw [Prod.norm_def,Real.norm_eq_abs]
exact max_le_add_of_nonneg (norm_nonneg x) (abs_nonneg y)
calc
_ ≤ ∫ y, ‖f (x,y)‖ ∂gaussianReal 0 1 := norm_integral_le_integral_norm _
_ ≤ ∫ y, C*Real.exp (M*‖x‖)*Real.exp (M*|y|) ∂gaussianReal 0 1 :=
integral_mono_of_nonneg (ae_of_all _ (fun _ => norm_nonneg _))
((integrable_exp_mul_abs_gaussian M).const_mul _) (ae_of_all _ hb)
_ = _ := by rw [integral_const_mul]; dsimp [B]; ring
theorem gaussianAverage_exp_representation {f : E × ℝ → ℝ} (hf : BoundedDerivs f)
(m : ℝ) (g : E × ℝ → F) (x : E) :
gaussianAverage m f g x = Real.exp (-m*gaussianStep m f x) •
∫ y, Real.exp (m*f (x,y)) • g (x,y) ∂gaussianReal 0 1 := by
rw [gaussianAverage,gaussianStepLaw_integral hf,← integral_smul]
apply integral_congr_ae
filter_upwards [] with y
rw [smul_smul,← Real.exp_add]
congr 2
ring
theorem gaussianAverage_expGrowth {f : E × ℝ → ℝ} (hf : BoundedDerivs f)
(m : ℝ) {g : E × ℝ → F} (hg : HasExpGrowth g) :
HasExpGrowth (gaussianAverage m f g) := by
have hn := (((hf.exp_growths m).1).smul hg).gaussian_integral
have hi := ((hf.gaussianStep m).exp_growths (-m)).1
rw [funext (gaussianAverage_exp_representation hf m g)]
exact hi.smul hn
theorem gaussianAverage_continuous_of_expGrowth {f : E × ℝ → ℝ} (hf : BoundedDerivs f)
(m : ℝ) {g : E × ℝ → F} (hg : HasExpGrowth g) (hc : Continuous g) :
Continuous (gaussianAverage m f g) := by
have he := (hf.exp_growths m).1
have hce : Continuous (fun z => Real.exp (m*f z)) :=
Real.continuous_exp.comp (continuous_const.mul hf.1.continuous)
have hn := (he.smul hg).continuous_gaussian_integral (hce.smul hc)
have hi : Continuous (fun x => Real.exp (-m*gaussianStep m f x)) :=
Real.continuous_exp.comp (continuous_const.mul (hf.gaussianStep m).1.continuous)
rw [funext (gaussianAverage_exp_representation hf m g)]
exact hi.smul hn
end ExponentialAverage
attribute [local instance 2000] parameterNormedGroup parameterNormedSpace
theorem hierarchyPathWeight_expGrowth (n : ℕ) (m : Fin n → ℝ)
(f : ParameterSpace n → ℝ) (hf : BoundedDerivs f) :
HasExpGrowth (hierarchyPathWeight n m f) := by
have hh := hf.add ((hierarchyPenalty_boundedDerivs n m 1 f hf).const_mul (-1))
have h := (hh.exp_growths 1).1
rw [funext (hierarchyPathWeight_eq n m f 1)]
simpa only [one_mul,neg_one_mul,← sub_eq_add_neg] using h
theorem hierarchyPath_exp_integrable (n : ℕ) (m : Fin n → ℝ)
(f : ParameterSpace n → ℝ) (hf : BoundedDerivs f) (x : ℝ)
(g : ParameterSpace n → ℝ) (hg : Continuous g) (hb : HasExpGrowth g) :
Integrable (fun z => hierarchyPathWeight n m f z*g z) (fiberGaussian n x) :=
((hierarchyPathWeight_expGrowth n m f hf).mul hb).integrable_fiberGaussian n
((hierarchyPathWeight_continuous n m f hf).mul hg) x
end SK.Analytic
end
end
end
end OAI
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