| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianAverage |
|
|
| 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 |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology NNReal |
| namespace SK.Analytic |
|
|
| @[reducible] def CascadeSpace (E : Type) : ℕ → Type |
| | 0 => E |
| | n+1 => CascadeSpace E n × ℝ |
| |
| instance cascadeNormedGroup (E : Type) [NormedAddCommGroup E] : |
| (n : ℕ) → NormedAddCommGroup (CascadeSpace E n) |
| | 0 => inferInstanceAs (NormedAddCommGroup E) |
| | n+1 => letI := cascadeNormedGroup E n |
| inferInstanceAs (NormedAddCommGroup (CascadeSpace E n × ℝ)) |
| |
| instance cascadeNormedSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : |
| (n : ℕ) → NormedSpace ℝ (CascadeSpace E n) |
| | 0 => inferInstanceAs (NormedSpace ℝ E) |
| | n+1 => letI := cascadeNormedSpace E n |
| inferInstanceAs (NormedSpace ℝ (CascadeSpace E n × ℝ)) |
| |
| section Cascade |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| |
| def cascadeLift : (n : ℕ) → E → CascadeSpace E n |
| | 0,u => u |
| | n+1,u => (cascadeLift n u,0) |
| |
| def cascadePressure : (n : ℕ) → (Fin n → ℝ) → (CascadeSpace E n → ℝ) → E → ℝ |
| | 0,_,f => f |
| | n+1,m,f => cascadePressure n (fun i => m i.castSucc) (gaussianStep (m (Fin.last n)) f) |
| |
| def cascadeMoment : (n : ℕ) → (Fin n → ℝ) → (CascadeSpace E n → ℝ) → |
| (CascadeSpace E n → ℝ) → E → ℝ |
| | 0,_,_,g => g |
| | n+1,m,f,g => cascadeMoment n (fun i => m i.castSucc) (gaussianStep (m (Fin.last n)) f) |
| (gaussianAverage (m (Fin.last n)) f g) |
| |
| theorem cascadePressure_boundedDerivs (n : ℕ) (m : Fin n → ℝ) |
| (f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) : BoundedDerivs (cascadePressure n m f) := by |
| induction n with |
| | zero => exact hf |
| | succ n ih => exact ih (fun i => m i.castSucc) _ (hf.gaussianStep _) |
| |
| theorem cascadeMoment_bounded_continuous (n : ℕ) (m : Fin n → ℝ) |
| (f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) (g : CascadeSpace E n → ℝ) |
| (hg : Continuous g) {C : ℝ} (hC : 0 ≤ C) (hb : ∀ z, ‖g z‖ ≤ C) : |
| Continuous (cascadeMoment n m f g) ∧ ∀ x, ‖cascadeMoment n m f g x‖ ≤ C := by |
| induction n with |
| | zero => exact ⟨hg,hb⟩ |
| | succ n ih => |
| exact ih (fun i => m i.castSucc) _ (hf.gaussianStep _) _ |
| (gaussianAverage_continuous hf _ hg hC hb) (fun x => gaussianAverage_norm_le hf _ hb x) |
| |
| theorem cascadePressure_gradient (n : ℕ) (m : Fin n → ℝ) |
| (f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) (x u : E) : |
| fderiv ℝ (cascadePressure n m f) x u = |
| cascadeMoment n m f (fun z => fderiv ℝ f z (cascadeLift n u)) x := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| rw [cascadePressure,ih (fun i => m i.castSucc) _ (hf.gaussianStep _)] |
| simp_rw [fderiv_gaussianStep_apply hf] |
| rfl |
| |
| theorem cascade_curvature_domination (n : ℕ) (m : Fin n → ℝ) |
| (f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) (A : CascadeSpace E n → ℝ) |
| (hA : Continuous A) {C upper lower : ℝ} (hC : 0 ≤ C) (hb : ∀ z, ‖A z‖ ≤ C) |
| (hmlu : lower ≤ upper) (hm : ∀ i, lower ≤ m i) (hmu : ∀ i, m i ≤ upper) |
| (hmono : Monotone m) (u : E) |
| (hV : ∀ z, 0 ≤ A z-(fderiv ℝ f z (cascadeLift n u))^2) |
| (hH : ∀ z, upper*(A z-(fderiv ℝ f z (cascadeLift n u))^2) ≤ |
| fderiv ℝ (fderiv ℝ f) z (cascadeLift n u) (cascadeLift n u)) (x : E) : |
| 0 ≤ cascadeMoment n m f A x-(fderiv ℝ (cascadePressure n m f) x u)^2 ∧ |
| lower*(cascadeMoment n m f A x-(fderiv ℝ (cascadePressure n m f) x u)^2) ≤ |
| fderiv ℝ (fderiv ℝ (cascadePressure n m f)) x u u := by |
| induction n generalizing upper with |
| | zero => |
| refine ⟨hV x,?_⟩ |
| exact (mul_le_mul_of_nonneg_right hmlu (hV x)).trans (hH x) |
| | succ n ih => |
| let q := m (Fin.last n) |
| let f' := gaussianStep q f |
| let A' := gaussianAverage q f A |
| have hv' (z : CascadeSpace E n) : 0 ≤ A' z-(fderiv ℝ f' z (cascadeLift n u))^2 := |
| gaussianStep_variance_nonneg hf hA hb q (cascadeLift n u) z (fun y => hV (z,y)) |
| have hh' (z : CascadeSpace E n) : |
| q*(A' z-(fderiv ℝ f' z (cascadeLift n u))^2) ≤ |
| fderiv ℝ (fderiv ℝ f') z (cascadeLift n u) (cascadeLift n u) := by |
| apply gaussianStep_curvature_domination hf hA hb q (cascadeLift n u) z |
| intro y |
| exact (mul_le_mul_of_nonneg_right (hmu (Fin.last n)) (hV (z,y))).trans (hH (z,y)) |
| exact ih (fun i => m i.castSucc) f' (hf.gaussianStep _) A' |
| (gaussianAverage_continuous hf _ hA hC hb) (fun z => gaussianAverage_norm_le hf _ hb z) |
| (hm (Fin.last n)) (fun i => hm i.castSucc) (fun i => hmono (Fin.le_last _)) |
| (fun i j hij => hmono (Fin.castSucc_le_castSucc_iff.mpr hij)) hv' hh' |
| |
| end Cascade |
| end SK.Analytic |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
