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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
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