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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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import Mathlib
import OAI.Probability.LogConcave.Sampling.CoordinateProjection
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open scoped Classical BigOperators
lemma Appell.HasGrowth.comp_linear {E F Z : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup Z]
{f : F → Z} (hf : Appell.HasGrowth f) (L : E →L[ℝ] F) :
Appell.HasGrowth (fun x => f (L x)) := by
obtain ⟨C,hC,n,hf⟩ := hf
refine ⟨C*(1+‖L‖^n),by positivity,n,fun x => (hf (L x)).trans ?_⟩
have hp : ‖L x‖^n≤‖L‖^n*‖x‖^n := by
simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _) (L.le_opNorm x) n
have hLn := pow_nonneg (norm_nonneg L) n
have hxn := pow_nonneg (norm_nonneg x) n
nlinarith [mul_le_mul_of_nonneg_left hp hC,mul_nonneg hC hLn,mul_nonneg hC hxn]
lemma PolySmooth.comp_linear {d e : ℕ} {f : Point e → ℝ}
(hf : PolySmooth f) (L : Point d →L[ℝ] Point e) : PolySmooth (fun x => f (L x)) := by
refine ⟨hf.smooth.comp L.contDiff,fun l => ?_⟩
have he := JetCalculus.jet_comp_affine hf.smooth L 0 id l
simp only [add_zero,id_eq] at he
rw [he]
have hh := hf.growth (l.map L)
rw [JetCalculus.jet_map] at hh
exact Appell.HasGrowth.comp_linear hh L
lemma productPotential_polySmooth {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
(hH : PolySmooth H) (hG : PolySmooth G) : PolySmooth (productPotential H G) :=
(hH.comp_linear ((ContinuousLinearMap.fst ℝ (Point d) (Point e)).comp
(productPointEquiv d e).toContinuousLinearMap)).add
(hG.comp_linear ((ContinuousLinearMap.snd ℝ (Point d) (Point e)).comp
(productPointEquiv d e).toContinuousLinearMap))
lemma gaussianPotential_polySmooth (d : ℕ) : PolySmooth (fun y : Point d => ‖y‖^2/2) := by
have h (i : Fin d) : PolySmooth (fun y : Point d => y i) := PolySmooth.linear (PiLp.proj 2 _ i)
have hs := PolySmooth.sum Finset.univ (fun i _ => (h i).mul (h i))
have hh := (PolySmooth.const (1/2)).mul hs
have he : (fun y : Point d => ‖y‖^2/2)=(fun y : Point d => (1/2 : ℝ)*(∑i,y i*y i)) := by
funext y
rw [EuclideanSpace.real_norm_sq_eq]
simp only [pow_two]
ring
rw [he]
exact hh
lemma productPotential_lowerTail {d e : ℕ} {H : Point d → ℝ} {G : Point e → ℝ}
(hH : HasGaussianLowerTail H) (hG : HasGaussianLowerTail G) :
HasGaussianLowerTail (productPotential H G) := by
obtain ⟨a,ha,A,hA⟩ := hH
obtain ⟨b,hb,B,hB⟩ := hG
refine ⟨min a b,lt_min ha hb,A+B,fun y => ?_⟩
rw [productPointEquiv_norm_sq]
have h₁ := mul_le_mul_of_nonneg_right (min_le_left a b) (sq_nonneg ‖(productPointEquiv d e y).1‖)
have h₂ := mul_le_mul_of_nonneg_right (min_le_right a b) (sq_nonneg ‖(productPointEquiv d e y).2‖)
have h₃ := hA (productPointEquiv d e y).1
have h₄ := hB (productPointEquiv d e y).2
dsimp [productPotential]
nlinarith
lemma gaussianPotential_lowerTail (d : ℕ) : HasGaussianLowerTail (fun y : Point d => ‖y‖^2/2) := by
exact ⟨1/2,by norm_num,0,fun _ => by linarith⟩
end LogConcaveSampling
end
end
end
end OAI
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