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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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Probability.LogConcave.Sampling.ConditionalMeanArray
import OAI.Probability.LogConcave.Sampling.InnerBasisCoordinate
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open scoped Classical BigOperators RealInnerProductSpace
open TensorEnergy
def harmonicKernel (d : ℕ) : Point d → Point d →L[ℝ] Point d :=
fun _ => -(ContinuousLinearMap.id ℝ (Point d))
def harmonicSkew (d : ℕ) := jointSkew (harmonicKernel d) 1
lemma harmonicKernel_strong {d : ℕ} (H : Point d → ℝ) (u : Point d) :
tensorAdjoint H (EuclideanSpace.basisFun (Fin d) ℝ)
(fun i y => inner ℝ u (harmonicKernel d y (EuclideanSpace.basisFun (Fin d) ℝ i)))=
fun y => inner ℝ u (-gradient H y) := by
funext y
simp only [tensorAdjoint,adjointCoordinate,harmonicKernel,neg_apply,
ContinuousLinearMap.id_apply,inner_neg_right]
simp only [directional,fderiv_fun_const,Pi.zero_apply,zero_apply,neg_zero,zero_add]
change (∑i : Fin d,directional (EuclideanSpace.basisFun (Fin d) ℝ i) H y*
(-inner ℝ u (EuclideanSpace.basisFun (Fin d) ℝ i)))= -inner ℝ u (gradient H y)
have hcu (i : Fin d) : inner ℝ u (EuclideanSpace.basisFun (Fin d) ℝ i)=u i := by
rw [real_inner_comm]
exact inner_basis_coordinate u i
simp only [directional_basis_gradient,hcu,mul_neg,Finset.sum_neg_distrib]
congr 1
lemma harmonicSkew_field {d : ℕ} {H : Point d → ℝ} (hH : Differentiable ℝ H)
(y : Point (d+d)) :
productPointEquiv d d (skewLieField (productPotential H (fun z : Point d => ‖z‖^2/2))
(harmonicSkew d) y)=((productPointEquiv d d y).2,-gradient H (productPointEquiv d d y).1) := by
have ha (z : Point d) : (harmonicKernel d z).adjoint=-(ContinuousLinearMap.id ℝ (Point d)) := by
change ContinuousLinearMap.adjoint (-(ContinuousLinearMap.id ℝ (Point d)))=_
rw [map_neg,ContinuousLinearMap.adjoint_id]
have h := jointSkew_field (K:=harmonicKernel d) (m:=fun z => -gradient H z)
hH (fun _ _ => by simpa only [harmonicKernel] using (differentiable_const _)) (harmonicKernel_strong H) 1 y
simpa only [harmonicSkew,skewCenteringField,ha,inv_one,one_smul,neg_smul,
neg_apply,ContinuousLinearMap.id_apply,neg_neg] using h
end LogConcaveSampling
end
end
section
noncomputable section
namespace LogConcaveSampling
open scoped Classical BigOperators RealInnerProductSpace
open TensorEnergy
lemma harmonicKernel_spatial_bound {d n : ℕ} (y : Point d) :
AllSplitBound (spatialTensor (fun c z => matrixArray d (harmonicKernel d z) c)
(List.finRange n) y) 1 := by
have hh := ((allSplitBound_spatial_delta (d:=d) (List.finRange n) (by simp)).reindex
(Equiv.sumAssoc (Fin n) Unit Unit)).abs_const_mul (-1)
have he : spatialTensor (fun c z => matrixArray d (harmonicKernel d z) c) (List.finRange n) y=
fun c => (-1:ℝ)*(if List.finRange n=[] then
(if c (.inr (.inl ()))=c (.inr (.inr ())) then (1:ℝ) else 0) else 0) := by
funext c
unfold spatialTensor
simp only [matrixArray_apply,harmonicKernel,neg_apply,ContinuousLinearMap.id_apply,
inner_neg_right,EuclideanSpace.basisFun_inner,JetCalculus.jet_const]
by_cases hn : List.finRange n=[] <;> simp [hn,EuclideanSpace.basisFun_apply,eq_comm]
split_ifs <;> norm_num
rw [he]
apply hh.mono
· split_ifs <;> norm_num
· split_ifs <;> norm_num
lemma harmonicSkew_polySmooth (d : ℕ) : ∀c,PolySmooth (harmonicSkew d c) :=
jointSkew_polySmooth (fun _ _ => PolySmooth.const _) 1
lemma harmonicSkew_scaled_bound (d : ℕ) : ∀k y,
AllSplitBound (spatialTensor (harmonicSkew d) (List.finRange k) y) ((2:ℝ)*1*1^k) := by
intro k y
simpa only [harmonicSkew,inv_one,abs_one,mul_one,one_pow] using
jointSkew_spatial_bound (n:=k) (K:=harmonicKernel d) (fun _ => by simpa only [harmonicKernel] using (contDiff_const (c:=_)))
(by norm_num : (0:ℝ)≤1) harmonicKernel_spatial_bound 1 y
end LogConcaveSampling
end
end
end
end OAI
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