| 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 PolyC1.of_lipschitz {d : ℕ} {f : Point d → ℝ} {K : ℝ≥0} |
| (hf : ContDiff ℝ 1 f) (hL : LipschitzWith K f) : PolyC1 f := by |
| refine ⟨hf,growth_of_lipschitz hL,fun v => ⟨(K:ℝ)*‖v‖,by positivity,0,fun x => ?_⟩⟩ |
| have hn := (fderiv ℝ f x).le_opNorm v |
| have hb := norm_fderiv_le_of_lipschitz ℝ hL (x₀:=x) |
| change ‖(fderiv ℝ f x) v‖ ≤ _ |
| simp only [pow_zero] |
| exact hn.trans ((mul_le_mul_of_nonneg_right hb (norm_nonneg v)).trans (by nlinarith [mul_nonneg K.coe_nonneg (norm_nonneg v)])) |
|
|
| lemma polyC1_directional_potential {d : ℕ} {H : Point d → ℝ} {K : ℝ≥0} |
| (hH : ContDiff ℝ (⊤ : ℕ∞) H) (hL : LipschitzWith K (gradient H)) (v : Point d) : |
| PolyC1 (directional v H) := by |
| apply PolyC1.of_lipschitz ((directional_smooth hH v).of_le (by simp)) |
| have he : (fun x => (innerSL ℝ v) (gradient H x))=directional v H := by |
| funext x |
| rw [innerSL_apply_apply,real_inner_comm,directional_gradient] |
| simpa only [Function.comp_def,he] using (innerSL ℝ v).lipschitzWith.comp hL |
|
|
| lemma polyC1_adjointCoordinate {d : ℕ} {H f : Point d → ℝ} {K : ℝ≥0} |
| (hH : ContDiff ℝ (⊤ : ℕ∞) H) (hL : LipschitzWith K (gradient H)) |
| (hf : PolyC1 f) (v : Point d) (hd : PolyC1 (directional v f)) : |
| PolyC1 (adjointCoordinate H v f) := |
| hd.neg.add ((polyC1_directional_potential hH hL v).mul hf) |
|
|
| lemma polyC1_tensorAdjoint {d : ℕ} {ι : Type*} [Fintype ι] |
| {H : Point d → ℝ} {V : ι → Point d → ℝ} {K : ℝ≥0} |
| (hH : ContDiff ℝ (⊤ : ℕ∞) H) (hL : LipschitzWith K (gradient H)) |
| (b : ι → Point d) (hV : ∀i,PolyC1 (V i)) (hD : ∀i,PolyC1 (directional (b i) (V i))) : |
| PolyC1 (tensorAdjoint H b V) := |
| PolyC1.sum Finset.univ (fun i _ => polyC1_adjointCoordinate hH hL (hV i) (b i) (hD i)) |
|
|
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
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|
