| import Mathlib |
| import OAI.Probability.LogConcave.Model |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| namespace LogConcaveSampling |
| open scoped RealInnerProductSpace |
| open MeasureTheory |
|
|
| def directional {d : ℕ} (v : Point d) (f : Point d → ℝ) : Point d → ℝ := |
| fun x => fderiv ℝ f x v |
|
|
| lemma directional_smooth {d : ℕ} {f : Point d → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (v : Point d) : |
| ContDiff ℝ (⊤ : ℕ∞) (directional v f) := |
| (hf.fderiv_right (by simp)).clm_apply contDiff_const |
|
|
| lemma directional_mul {d : ℕ} {f g : Point d → ℝ} (hf : Differentiable ℝ f) |
| (hg : Differentiable ℝ g) (v : Point d) : |
| directional v (fun x => f x*g x)=fun x => directional v f x*g x+f x*directional v g x := by |
| funext x |
| unfold directional |
| change (fderiv ℝ (f*g) x) v=_ |
| rw [((hf x).hasFDerivAt.mul (hg x).hasFDerivAt).fderiv] |
| simp only [add_apply,smul_apply,smul_eq_mul] |
| ring |
|
|
| lemma directional_add {d : ℕ} {f g : Point d → ℝ} (hf : Differentiable ℝ f) |
| (hg : Differentiable ℝ g) (v : Point d) : |
| directional v (fun x => f x+g x)=fun x => directional v f x+directional v g x := by |
| funext x |
| exact congrArg (fun A : Point d →L[ℝ] ℝ => A v) ((hf x).hasFDerivAt.add (hg x).hasFDerivAt).fderiv |
|
|
| lemma directional_neg {d : ℕ} {f : Point d → ℝ} (hf : Differentiable ℝ f) (v : Point d) : |
| directional v (fun x => -f x)=fun x => -directional v f x := by |
| funext x |
| exact congrArg (fun A : Point d →L[ℝ] ℝ => A v) ((hf x).hasFDerivAt.neg).fderiv |
|
|
| lemma directional_sum {d : ℕ} {ι : Type*} (s : Finset ι) {f : ι → Point d → ℝ} |
| (hf : ∀ i∈s,Differentiable ℝ (f i)) (v : Point d) : |
| directional v (fun x => ∑ i∈s,f i x)=fun x => ∑ i∈s,directional v (f i) x := by |
| funext x |
| have hh := HasFDerivAt.fun_sum (u:=s) (fun i hi => (hf i hi x).hasFDerivAt) |
| simpa only [directional,sum_apply] using |
| congrArg (fun A : Point d →L[ℝ] ℝ => A v) hh.fderiv |
|
|
| lemma directional_directional {d : ℕ} {f : Point d → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (u v x : Point d) : |
| directional u (directional v f) x=(fderiv ℝ (fderiv ℝ f) x u) v := by |
| have hdf : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp) |
| unfold directional |
| rw [fderiv_clm_apply (hdf.differentiable (by norm_num) x) (differentiableAt_const _)] |
| simp |
|
|
| lemma directional_commute {d : ℕ} {f : Point d → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (u v : Point d) : |
| directional u (directional v f)=directional v (directional u f) := by |
| funext x |
| rw [directional_directional hf,directional_directional hf] |
| exact ((hf.contDiffAt (x:=x)).isSymmSndFDerivAt (by |
| simp only [minSmoothness_of_isRCLikeNormedField] |
| exact WithTop.coe_le_coe.mpr (show (2:ℕ∞)≤⊤ from le_top))).eq u v |
|
|
| lemma directional_gradient {d : ℕ} (H : Point d → ℝ) (v x : Point d) : |
| directional v H x=inner ℝ (gradient H x) v := by |
| rw [directional,←toDual_gradient,InnerProductSpace.toDual_apply_apply] |
|
|
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
