openAI-math-dataset / lean /OAI /Probability /LogConcave /Sampling /DirectionalAdjointCoordinate.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.JetEstimates.PolyC1 | |
| 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 MeasureTheory | |
| open scoped RealInnerProductSpace | |
| lemma directional_adjointCoordinate {d : ℕ} {H f : Point d → ℝ} | |
| (hH : ContDiff ℝ (⊤ : ℕ∞) H) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (u v : Point d) : | |
| directional u (adjointCoordinate H v f)=fun x => | |
| adjointCoordinate H v (directional u f) x+directional u (directional v H) x*f x := by | |
| have hD : Differentiable ℝ (directional v f) := (directional_smooth hf v).differentiable (by simp) | |
| have hS : Differentiable ℝ (directional v H) := (directional_smooth hH v).differentiable (by simp) | |
| change directional u (fun x => -directional v f x+directional v H x*f x)=_ | |
| rw [directional_add (f:=fun x => -directional v f x) | |
| (g:=fun x => directional v H x*f x) hD.neg (hS.mul (hf.differentiable (by simp))),directional_neg hD, | |
| directional_mul hS (hf.differentiable (by simp)),directional_commute hf u v] | |
| funext x | |
| simp only [adjointCoordinate] | |
| ring | |
| lemma adjointCoordinate_smooth {d : ℕ} {H f : Point d → ℝ} | |
| (hH : ContDiff ℝ (⊤ : ℕ∞) H) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (v : Point d) : | |
| ContDiff ℝ (⊤ : ℕ∞) (adjointCoordinate H v f) := | |
| (directional_smooth hf v).neg.add ((directional_smooth hH v).mul hf) | |
| lemma directional_tensorAdjoint {d : ℕ} {ι : Type*} [Fintype ι] | |
| {H : Point d → ℝ} {V : ι → Point d → ℝ} | |
| (hH : ContDiff ℝ (⊤ : ℕ∞) H) (hV : ∀i,ContDiff ℝ (⊤ : ℕ∞) (V i)) | |
| (b : ι → Point d) (u : Point d) : | |
| directional u (tensorAdjoint H b V)=fun x => | |
| tensorAdjoint H b (fun i => directional u (V i)) x+ | |
| ∑i,directional u (directional (b i) H) x*V i x := by | |
| unfold tensorAdjoint | |
| rw [directional_sum Finset.univ | |
| (fun i _ => (adjointCoordinate_smooth hH (hV i) (b i)).differentiable (by simp))] | |
| simp_rw [directional_adjointCoordinate hH (hV _)] | |
| funext x | |
| exact Finset.sum_add_distrib | |
| end LogConcaveSampling | |
| end UpperProof | |
| end | |
| end | |
| end | |
| end OAI | |