| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.CubeMoments |
| import OAI.Probability.SKBarriers.Calculus.CoordinateBounds |
|
|
| 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 |
| namespace SK.Analytic |
|
|
| theorem stronglyConvex_linear_variance_le (n : ℕ) (V : ParameterSpace n → ℝ) |
| (hV : ContDiff ℝ 2 V) (hV₀ : ∀ z, parameterDerivative n V z = 0) |
| (L : ParameterSpace n →L[ℝ] ℝ) (hL₀ : L (parameterAxis n) = 0) |
| {c A : ℝ} (hc : 0 < c) (hA : 0 ≤ A) |
| (hL : ∀ u, (L u)^2 ≤ A*coordinateSquare n u) |
| (hH : ∀ z u, c*coordinateSquare n u ≤ Hessian V z u u) : |
| let Z := ∫ z, Real.exp (-V z) ∂fiberMeasure n 0 |
| (∫ z, (L z)^2*Real.exp (-V z) ∂fiberMeasure n 0)/Z - |
| ((∫ z, L z*Real.exp (-V z) ∂fiberMeasure n 0)/Z)^2 ≤ A/c := by |
| obtain ⟨hI₀,hIq⟩ := integrable_stronglyConvex_density n V hV hc hH |
| have hI₂ : Integrable (fun z => (L z)^2*Real.exp (-V z)) (fiberMeasure n 0) := by |
| apply (hIq.const_mul A).mono' ((L.continuous.pow 2).mul hV.neg.exp.continuous).aestronglyMeasurable |
| filter_upwards [] with z |
| change ‖(L z)^2*Real.exp (-V z)‖ ≤ _ |
| rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (sq_nonneg _) (Real.exp_pos _).le)] |
| exact (mul_le_mul_of_nonneg_right (hL z) (Real.exp_pos _).le).trans_eq (mul_assoc _ _ _) |
| have hI₁ : Integrable (fun z => L z*Real.exp (-V z)) (fiberMeasure n 0) := by |
| apply (hI₀.add hI₂).mono' (L.continuous.mul hV.neg.exp.continuous).aestronglyMeasurable |
| filter_upwards [] with z |
| change ‖L z*Real.exp (-V z)‖ ≤ Real.exp (-V z)+(L z)^2*Real.exp (-V z) |
| rw [Real.norm_eq_abs, abs_mul, abs_of_pos (Real.exp_pos _)] |
| have hle : |L z| ≤ 1+(L z)^2 := by nlinarith [sq_nonneg (|L z|-1), sq_abs (L z)] |
| nlinarith [mul_le_mul_of_nonneg_right hle (Real.exp_pos (-V z)).le] |
| exact linear_variance_le n V hV hV₀ L hL₀ hc hA hL hH hI₀ hI₁ hI₂ |
|
|
| section HessianComposition |
| variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| theorem hessian_comp_linear (V : F → ℝ) (hV : ContDiff ℝ 2 V) (T : E →L[ℝ] F) |
| (x u v : E) : Hessian (fun z => V (T z)) x u v = Hessian V (T x) (T u) (T v) := by |
| have hD : fderiv ℝ (fun z => V (T z)) = fun z => (fderiv ℝ V (T z)).comp T := by |
| funext z |
| exact ((hV.differentiable (by norm_num) (T z)).hasFDerivAt.comp z T.hasFDerivAt).fderiv |
| have hdV := (hV.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) |
| have hd := ((hdV (T x)).hasFDerivAt.comp x T.hasFDerivAt).clm_comp (hasFDerivAt_const T x) |
| simp only [Function.comp_def] at hd |
| rw [Hessian, hD, hd.fderiv] |
| simp [Hessian] |
|
|
| theorem hessian_linear (L : E →L[ℝ] ℝ) (x u v : E) : Hessian L x u v = 0 := by |
| have hD : fderiv ℝ (fun z => L z) = fun _ => L := funext (fun _ => L.hasFDerivAt.fderiv) |
| change Hessian (fun z => L z) x u v = 0 |
| simp [Hessian, hD] |
|
|
| end HessianComposition |
|
|
| theorem contDiff_coordinateSquare (n : ℕ) : ContDiff ℝ 2 (coordinateSquare n) := by |
| induction n with |
| | zero => exact contDiff_const |
| | succ n ih => exact (ih.comp contDiff_fst).add (contDiff_snd.pow 2) |
|
|
| theorem hessian_coordinateSquare (n : ℕ) (x u : ParameterSpace n) : |
| Hessian (coordinateSquare n) x u u = 2*coordinateSquare n u := by |
| induction n with |
| | zero => simp [coordinateSquare, Hessian] |
| | succ n ih => |
| let fst := ContinuousLinearMap.fst ℝ (ParameterSpace n) ℝ |
| let snd := ContinuousLinearMap.snd ℝ (ParameterSpace n) ℝ |
| change Hessian (fun z => coordinateSquare n (fst z)+(snd z)^2) x u u = _ |
| have hf : ContDiff ℝ 2 (fun z => coordinateSquare n (fst z)) := (contDiff_coordinateSquare n).comp fst.contDiff |
| rw [hessian_add hf (snd.contDiff.pow 2), |
| hessian_comp_linear _ (contDiff_coordinateSquare n) fst] |
| have hsq : (fun z => (snd z)^2) = fun z => snd z*snd z := by funext z; ring |
| rw [hsq, hessian_linear_product, ih] |
| change 2*coordinateSquare n u.1 + (u.2*u.2+u.2*u.2) = 2*(coordinateSquare n u.1+u.2^2) |
| ring |
|
|
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
