| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.LocalGaussian |
|
|
| namespace OAI |
|
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| section |
|
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
|
|
| def ParamExpGrowth {P E F : Type} [Norm P] [Norm E] [Norm F] |
| (f : P × E → F) : Prop := |
| ∀ R : ℝ, ∃ C M : ℝ, 0 ≤ C ∧ 0 ≤ M ∧ |
| ∀ p e, ‖p‖ ≤ R → ‖f (p,e)‖ ≤ C*Real.exp (M*‖e‖) |
|
|
| def ParamLinearGrowth {P E : Type} [Norm P] [Norm E] |
| (f : P × E → ℝ) : Prop := |
| ∀ R : ℝ, ∃ C : ℝ, 0 ≤ C ∧ |
| ∀ p e, ‖p‖ ≤ R → |f (p,e)| ≤ C*(1+‖e‖) |
|
|
| theorem ParamExpGrowth.of_norm_le {P E F G : Type} [Norm P] [Norm E] [Norm F] [Norm G] |
| {f : P × E → F} (hf : ParamExpGrowth f) |
| {g : P × E → G} {K : ℝ} (hK : 0 ≤ K) (hg : ∀ z, ‖g z‖ ≤ K*‖f z‖) : |
| ParamExpGrowth g := by |
| intro R |
| obtain ⟨C,M,hC,hM,hf⟩ := hf R |
| refine ⟨K*C,M,mul_nonneg hK hC,hM,?_⟩ |
| intro p e hp |
| exact (hg (p,e)).trans ((mul_le_mul_of_nonneg_left (hf p e hp) hK).trans_eq (mul_assoc _ _ _).symm) |
|
|
| section ParameterGrowth |
| variable {P E F G : Type} [NormedAddCommGroup P] [NormedAddCommGroup E] |
| [NormedAddCommGroup F] [NormedAddCommGroup G] |
|
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| theorem ParamExpGrowth.of_bounded {f : P × E → F} {C : ℝ} (hC : 0 ≤ C) |
| (hf : ∀ z, ‖f z‖ ≤ C) : ParamExpGrowth f := by |
| intro R |
| refine ⟨C,0,hC,le_rfl,?_⟩ |
| intro p e _ |
| simpa only [zero_mul,Real.exp_zero,mul_one] using hf (p,e) |
|
|
| theorem ParamExpGrowth.add {f g : P × E → F} (hf : ParamExpGrowth f) |
| (hg : ParamExpGrowth g) : ParamExpGrowth (fun z => f z+g z) := by |
| intro R |
| obtain ⟨C,M,hC,hM,hf⟩ := hf R |
| obtain ⟨D,L,hD,hL,hg⟩ := hg R |
| refine ⟨C+D,max M L,add_nonneg hC hD,(le_max_left M L).trans' hM,?_⟩ |
| intro p e hp |
| calc |
| _ ≤ ‖f (p,e)‖+‖g (p,e)‖ := norm_add_le _ _ |
| _ ≤ C*Real.exp (M*‖e‖)+D*Real.exp (L*‖e‖) := add_le_add (hf p e hp) (hg p e hp) |
| _ ≤ C*Real.exp (max M L*‖e‖)+D*Real.exp (max M L*‖e‖) := by |
| gcongr <;> first | exact le_max_left _ _ | exact le_max_right _ _ |
| _ = _ := by ring |
|
|
| theorem ParamExpGrowth.mul {f g : P × E → ℝ} (hf : ParamExpGrowth f) |
| (hg : ParamExpGrowth g) : ParamExpGrowth (fun z => f z*g z) := by |
| intro R |
| obtain ⟨C,M,hC,hM,hf⟩ := hf R |
| obtain ⟨D,L,hD,hL,hg⟩ := hg R |
| refine ⟨C*D,M+L,mul_nonneg hC hD,add_nonneg hM hL,?_⟩ |
| intro p e hp |
| calc |
| ‖f (p,e)*g (p,e)‖ = ‖f (p,e)‖*‖g (p,e)‖ := norm_mul _ _ |
| _ ≤ (C*Real.exp (M*‖e‖))*(D*Real.exp (L*‖e‖)) := |
| mul_le_mul (hf p e hp) (hg p e hp) (norm_nonneg _) (by positivity) |
| _ = _ := by rw [add_mul,Real.exp_add]; ring |
|
|
| theorem ParamExpGrowth.norm {f : P × E → F} (hf : ParamExpGrowth f) : |
| ParamExpGrowth (fun z => ‖f z‖) := by |
| apply hf.of_norm_le (K := 1) (by norm_num) |
| intro z |
| simp |
|
|
| theorem ParamLinearGrowth.exp_mul {f : P × E → ℝ} (hf : ParamLinearGrowth f) (m : ℝ) : |
| ParamExpGrowth (fun z => Real.exp (m*f z)) := by |
| intro R |
| obtain ⟨C,hC,hf⟩ := hf R |
| refine ⟨Real.exp (|m| *C),|m| *C,(Real.exp_pos _).le,mul_nonneg (abs_nonneg _) hC,?_⟩ |
| intro p e hp |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),← Real.exp_add] |
| apply Real.exp_le_exp.2 |
| calc |
| m*f (p,e) ≤ |m*f (p,e)| := le_abs_self _ |
| _ = |m| *|f (p,e)| := abs_mul _ _ |
| _ ≤ |m| *(C*(1+‖e‖)) := mul_le_mul_of_nonneg_left (hf p e hp) (abs_nonneg m) |
| _ = _ := by ring |
|
|
| theorem ParamLinearGrowth.expGrowth {f : P × E → ℝ} (hf : ParamLinearGrowth f) : |
| ParamExpGrowth f := by |
| intro R |
| obtain ⟨C,hC,hf⟩ := hf R |
| refine ⟨C,1,hC,by norm_num,?_⟩ |
| intro p e hp |
| rw [one_mul,Real.norm_eq_abs] |
| exact (hf p e hp).trans (mul_le_mul_of_nonneg_left (by have h := Real.add_one_le_exp ‖e‖; linarith) hC) |
|
|
| theorem ParamExpGrowth.locallyDominated {f : P × (E × ℝ) → F} (hf : ParamExpGrowth f) : |
| GaussianLocallyDominated (fun z : (P × E) × ℝ => f (z.1.1,(z.1.2,z.2))) := by |
| intro R |
| obtain ⟨C,M,hC,hM,hf⟩ := hf R |
| refine ⟨fun y => C*Real.exp (M*R)*Real.exp (M*|y|), |
| (integrable_exp_mul_abs_gaussian M).const_mul _,?_⟩ |
| rintro ⟨p,e⟩ y hpe |
| have hp : ‖p‖ ≤ R := (le_max_left _ _).trans hpe |
| have he : ‖e‖ ≤ R := (le_max_right _ _).trans hpe |
| apply (hf p (e,y) hp).trans |
| change C*Real.exp (M*‖(e,y)‖) ≤ C*Real.exp (M*R)*Real.exp (M*|y|) |
| 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 (he.trans (le_add_of_nonneg_right (abs_nonneg _))) |
| (le_add_of_nonneg_left ((norm_nonneg e).trans he)) |
|
|
| variable [NormedSpace ℝ F] [NormedSpace ℝ G] |
|
|
| theorem ParamExpGrowth.clm {f : P × E → F} (hf : ParamExpGrowth f) (L : F →L[ℝ] G) : |
| ParamExpGrowth (fun z => L (f z)) := hf.of_norm_le (norm_nonneg L) (fun z => L.le_opNorm (f z)) |
|
|
| theorem ParamExpGrowth.smul {f : P × E → ℝ} {g : P × E → F} |
| (hf : ParamExpGrowth f) (hg : ParamExpGrowth g) : ParamExpGrowth (fun z => f z • g z) := by |
| apply (hf.norm.mul hg.norm).of_norm_le (K := 1) (by norm_num) |
| intro z |
| simp only [norm_smul,Real.norm_eq_abs,abs_mul,abs_abs,abs_of_nonneg (norm_nonneg _),one_mul] |
| exact le_rfl |
|
|
| theorem ParamExpGrowth.gaussian_integral {f : P × (E × ℝ) → F} |
| (hf : ParamExpGrowth f) : |
| ParamExpGrowth (fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1) := by |
| intro R |
| obtain ⟨C,M,hC,hM,hf⟩ := hf R |
| 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 p e hp |
| have hb : ∀ y, ‖f (p,(e,y))‖ ≤ C*Real.exp (M*‖e‖)*Real.exp (M*|y|) := by |
| intro y |
| apply (hf p (e,y) hp).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 e) (abs_nonneg y) |
| calc |
| _ ≤ ∫ y, ‖f (p,(e,y))‖ ∂gaussianReal 0 1 := norm_integral_le_integral_norm _ |
| _ ≤ ∫ y, C*Real.exp (M*‖e‖)*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 |
| end ParameterGrowth |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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