openAI-math-dataset / lean /OAI /Probability /LogConcave /JetEstimates /ArrayComponentPolySmooth.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Sampling.MatrixArrayAdjoint | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open scoped Classical BigOperators NNReal | |
| open TensorEnergy | |
| lemma arrayComponent_polySmooth {d : ℕ} {S : Type} [Fintype S] [DecidableEq S] | |
| (hS : 2≤Fintype.card S) {g : Point d → (S → Fin d) → ℝ} | |
| (hg : ContDiff ℝ (⊤:ℕ∞) g) (M : ℕ → ℝ) (hM : ∀n,0≤M n) | |
| (hB : ∀n y,AllSplitBound (arrayTensor (iteratedFDeriv ℝ n g y)) (M n)) | |
| (c : S → Fin d) : PolySmooth (fun y => g y c) := by | |
| apply PolySmooth.of_basis_growth ((ContinuousLinearMap.proj c).contDiff.comp hg) | |
| intro l | |
| apply growth_of_bounded (hM l.length) | |
| intro y | |
| have hh := (hB l.length y).reindex (Equiv.sumComm S (Fin l.length)) | |
| rw [←spatialTensor_eq_arrayTensor hg y] at hh | |
| have hb := hh.entry_le (hM _) (by simp only [Fintype.card_sum,Fintype.card_fin]; omega) | |
| (Sum.elim l.get c) | |
| change |JetCalculus.jet ((EuclideanSpace.basisFun (Fin d) ℝ) ∘ l.get) | |
| (List.finRange l.length) (fun y => g y c) y|≤_ at hb | |
| rw [←JetCalculus.jet_map,List.map_get_finRange] at hb | |
| exact hb | |
| lemma centeringKernel_polySmooth {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} | |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) | |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) | |
| (hT0 : 0≤T) (hT1 : T<1) (c : Unit ⊕ Unit → Fin d) : | |
| PolySmooth (fun y => matrixArray d (centeringKernel hF x hr hl hT0 hT1 y) c) := by | |
| let R := Real.sqrt (1-T^2) | |
| have hR : 0<R := Real.sqrt_pos.mpr (by nlinarith) | |
| have hRT : R^2≤1-T^2 := (Real.sq_sqrt (by nlinarith)).le | |
| exact arrayComponent_polySmooth (by simp) | |
| ((matrixArray d).contDiff.comp (centeringKernel_smooth hF x hr hlam hl hT0 hT1)) | |
| (fun n => ((lam:ℝ)*r)*kernelMajorant n/R^n) | |
| (fun n => div_nonneg (mul_nonneg (mul_nonneg lam.2 hr.le) (kernelMajorant_nonneg _)) (pow_nonneg hR.le _)) | |
| (fun _ y => centeringKernel_iterated_split hF x hr hlam hl hR hRT hT0 hT1 y) c | |
| end LogConcaveSampling | |
| end | |
| end | |
| end | |
| end OAI | |