| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.HierarchyDensity |
| import OAI.Probability.SKBarriers.Scalar.GrowthOperations |
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| namespace OAI |
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| section |
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| 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] |
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|
| 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 |
|
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| 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 |
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|
| end |
| end |
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| end |
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| end OAI |
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