| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianDomination |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology |
| namespace SK.Analytic |
| section UniformBounds |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| theorem norm_le_of_fderiv_bound (f : E → ℝ) (hd : Differentiable ℝ f) |
| (C : ℝ) (hC : ∀ x, ‖fderiv ℝ f x‖ ≤ C) (x : E) : |
| ‖f x‖ ≤ ‖f 0‖ + C*‖x‖ := by |
| have h := Convex.norm_image_sub_le_of_norm_fderiv_le |
| (fun x (_ : x ∈ (Set.univ : Set E)) => hd x) |
| (fun x (_ : x ∈ (Set.univ : Set E)) => hC x) convex_univ |
| (mem_univ (0:E)) (mem_univ x) |
| simp only [sub_zero] at h |
| calc |
| ‖f x‖ = ‖(f x-f 0)+f 0‖ := by rw [sub_add_cancel] |
| _ ≤ ‖f x-f 0‖+‖f 0‖ := norm_add_le _ _ |
| _ ≤ ‖f 0‖+C*‖x‖ := by linarith |
|
|
| theorem exp_mul_growth_of_fderiv_bound (f : E → ℝ) (hd : Differentiable ℝ f) |
| (C : ℝ) (hC₀ : 0 ≤ C) (hC : ∀ x, ‖fderiv ℝ f x‖ ≤ C) (m : ℝ) : |
| HasExpGrowth (fun x => Real.exp (m*f x)) := by |
| refine ⟨Real.exp (|m| *‖f 0‖), |m| *C, (Real.exp_pos _).le, |
| mul_nonneg (abs_nonneg _) hC₀, ?_⟩ |
| intro x |
| rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _), ← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have hf := norm_le_of_fderiv_bound f hd C hC x |
| have hb : m*f x ≤ |m| *‖f x‖ := by |
| simpa only [Real.norm_eq_abs, abs_mul] using le_abs_self (m*f x) |
| nlinarith [mul_le_mul_of_nonneg_left hf (abs_nonneg m)] |
|
|
| def positiveGaussianLogStep (m : ℝ) (f : E × ℝ → ℝ) (x : E) : ℝ := |
| Real.log (∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1) / m |
|
|
| theorem positiveGaussianLogStep_partition_pos (f : E × ℝ → ℝ) |
| (hd : Differentiable ℝ f) (C : ℝ) (hC₀ : 0 ≤ C) |
| (hC : ∀ x, ‖fderiv ℝ f x‖ ≤ C) (m : ℝ) (x : E) : |
| 0 < ∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1 := by |
| apply integral_exp_pos |
| exact (exp_mul_growth_of_fderiv_bound f hd C hC₀ hC m).integrable_gaussian_section |
| (Real.continuous_exp.comp (continuous_const.mul hd.continuous)) x |
|
|
| end UniformBounds |
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
