| 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 |
|
|
