| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.SpaceBasis |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
|
|
| lemma tensorAdjoint_scalar {d : ℕ} {ι : Type*} [Fintype ι] |
| (H : Point d → ℝ) (b : ι → Point d) {V : ι → Point d → ℝ} |
| (hV : ∀i,Differentiable ℝ (V i)) (c : ℝ) : |
| tensorAdjoint H b (fun i y => c*V i y)=fun y => c*tensorAdjoint H b V y := by |
| unfold tensorAdjoint adjointCoordinate |
| change (fun y => ∑i, (-JetCalculus.dir (b i) (fun z => c*V i z) y+ |
| directional (b i) H y*(c*V i y)))=_ |
| simp_rw [JetCalculus.dir_const_mul (hV _)] |
| funext y |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| change -(c*JetCalculus.dir (b i) (V i) y)+directional (b i) H y*(c*V i y)= |
| c*(-JetCalculus.dir (b i) (V i) y+directional (b i) H y*V i y) |
| ring |
|
|
| lemma conditionalU_one_eq_dir {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) |
| (u v : Point d) : |
| (fun y => conditionalU F x r ρ y u ((lam:ℝ)*r) (fun _ : Unit => v) [()])= |
| fun y => ρ⁻¹*JetCalculus.dir v (fun z => inner ℝ u (conditionalFieldMean F x r ρ z)) y := by |
| funext y |
| have hh := conditionalFieldMean_jet_U hF x hr hlam hl hρ0.le hρ1 u y |
| (fun _ : Unit => v) [()] |
| have hm : conditionalMeanScalar F x r ρ (fun z => inner ℝ u (primitiveField F x r z))= |
| fun y => inner ℝ u (conditionalFieldMean F x r ρ y) := |
| (funext (conditionalFieldMean_inner hF x hr.le hl hρ0.le hρ1 u)).symm |
| rw [hm] at hh |
| simp only [JetCalculus.jet,List.length_singleton,pow_one] at hh |
| rw [hh] |
| field_simp [hρ0.ne'] |
|
|
| lemma material_score_algebra {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (u y : Point d) : |
| materialScalar F x r (fun t z => inner ℝ u z+r*t*inner ℝ u (conditionalFieldMean F x r t z)) ρ y= |
| r*ρ*materialScalar F x r (fun t z => inner ℝ u (conditionalFieldMean F x r t z)) ρ y := by |
| let m := fun t z => inner ℝ u (conditionalFieldMean F x r t z) |
| have hsq : ρ^2<1 := by have := (probability_time hρ0 hρ1).1; linarith |
| have hj : DifferentiableAt ℝ (fun p : ℝ × Point d => m p.1 p.2) (ρ,y) := |
| ((innerSL ℝ u).contDiff.contDiffAt.comp (ρ,y) |
| (conditionalFieldMean_joint_smooth hF x hr (by linarith) (ρ,y) hsq)).differentiableAt (by simp) |
| have hi : DifferentiableAt ℝ (fun t : ℝ => (t,y)) ρ := |
| differentiableAt_id.prodMk (differentiableAt_const y) |
| have ht : DifferentiableAt ℝ (fun t => m t y) ρ := by |
| have hh := hj.comp ρ hi |
| exact hh |
| have hs : Differentiable ℝ (m ρ) := |
| ((innerSL ℝ u).contDiff.comp (conditionalFieldMean_smooth hF x hr hl hρ0 hρ1)).differentiable (by simp) |
| have htd : deriv (fun t => inner ℝ u y+r*t*m t y) ρ= |
| r*m ρ y+(r*ρ)*deriv (fun t => m t y) ρ := by |
| have hh := (hasDerivAt_const ρ (inner ℝ u y)).add |
| (((hasDerivAt_id ρ).const_mul r).mul ht.hasDerivAt) |
| simpa only [Pi.add_def,Pi.mul_def,id_eq,mul_one,zero_add] using hh.deriv |
| have hsd : directional (conditionalFieldMean F x r ρ y) |
| (fun z => inner ℝ u z+r*ρ*m ρ z) y= |
| m ρ y+(r*ρ)*directional (conditionalFieldMean F x r ρ y) (m ρ) y := by |
| unfold directional |
| have hh := (innerSL ℝ u).hasFDerivAt.add ((hs y).hasFDerivAt.const_mul (r*ρ)) |
| change (fderiv ℝ (((innerSL ℝ u) : Point d → ℝ) + fun z => (r*ρ)*m ρ z) y) _=_ |
| rw [hh.fderiv] |
| simp [m] |
| change deriv (fun t => inner ℝ u y+r*t*m t y) ρ- |
| r*directional (conditionalFieldMean F x r ρ y) (fun z => inner ℝ u z+r*ρ*m ρ z) y= |
| r*ρ*(deriv (fun t => m t y) ρ-r*directional (conditionalFieldMean F x r ρ y) (m ρ) y) |
| rw [htd,hsd] |
| ring |
|
|
| theorem material_interpolation_score {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) (u y : Point d) : |
| materialScalar F x r (fun t z => inner ℝ u z+r*t*inner ℝ u (conditionalFieldMean F x r t z)) ρ y= |
| r*tensorAdjoint (interpolationPotential F x r ρ) (EuclideanSpace.basisFun (Fin d) ℝ) |
| (fun i => JetCalculus.dir u (fun z => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) |
| (conditionalFieldMean F x r ρ z))) y := by |
| rw [material_score_algebra hF x hr.le hl hρ0.le hρ1, |
| material_conditionalFieldMean hF x hr hlam hl hρ0.le hρ1] |
| have he : (fun i y => conditionalU F x r ρ y u ((lam:ℝ)*r) |
| (fun _ : Unit => EuclideanSpace.basisFun (Fin d) ℝ i) [()])= |
| fun i y => ρ⁻¹*JetCalculus.dir u (fun z => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) |
| (conditionalFieldMean F x r ρ z)) y := by |
| funext i y |
| rw [congrFun (conditionalU_one_eq_dir hF x hr hlam hl hρ0 hρ1 u _) y] |
| rw [conditionalFieldMean_directional_symm hF x hr hl hρ0 hρ1] |
| have hD (i : Fin d) : Differentiable ℝ (JetCalculus.dir u (fun z => |
| inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) (conditionalFieldMean F x r ρ z))) := |
| (JetCalculus.smooth_dir ((innerSL ℝ _).contDiff.comp |
| (conditionalFieldMean_smooth hF x hr.le hl hρ0.le hρ1)) u).differentiable (by simp) |
| rw [he,tensorAdjoint_scalar _ _ hD] |
| field_simp [hρ0.ne'] |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
|
|
| def jointSpace {d : ℕ} (i : Fin d) : ℝ × Point d := |
| (0,EuclideanSpace.basisFun (Fin d) ℝ i) |
|
|
| def jointMean {d : ℕ} (F : Point d → ℝ) (x : Point d) (r : ℝ) (i : Fin d) |
| (p : ℝ × Point d) : ℝ := |
| inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) (conditionalFieldMean F x r p.1 p.2) |
|
|
| def jointScore {d : ℕ} (F : Point d → ℝ) (x : Point d) (r : ℝ) (i : Fin d) |
| (p : ℝ × Point d) : ℝ := |
| inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) p.2+r*p.1*jointMean F x r i p |
|
|
| lemma jointMean_smooth_at {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) {p : ℝ × Point d} (hp : p.1^2<1) (i : Fin d) : |
| ContDiffAt ℝ (⊤:ℕ∞) (jointMean F x r i) p := |
| (innerSL ℝ _).contDiff.contDiffAt.comp p |
| (conditionalFieldMean_joint_smooth hF x hr (by linarith) p hp) |
|
|
| lemma jointScore_smooth_at {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) {p : ℝ × Point d} (hp : p.1^2<1) (i : Fin d) : |
| ContDiffAt ℝ (⊤:ℕ∞) (jointScore F x r i) p := |
| ((innerSL ℝ _).contDiff.contDiffAt.comp p contDiffAt_snd).add |
| ((contDiffAt_const.mul contDiffAt_fst).mul (jointMean_smooth_at hF x hr hl hp i)) |
|
|
| lemma joint_gadj_slice {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) |
| {V : Fin d → ℝ × Point d → ℝ} {y : Point d} |
| (hV : ∀i,DifferentiableAt ℝ (V i) (ρ,y)) : |
| JetCalculus.gadj jointSpace (jointScore F x r) V (ρ,y)= |
| tensorAdjoint (interpolationPotential F x r ρ) (EuclideanSpace.basisFun (Fin d) ℝ) |
| (fun i z => V i (ρ,z)) y := by |
| unfold JetCalculus.gadj JetCalculus.cadj tensorAdjoint adjointCoordinate |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp only [jointSpace] |
| rw [←JetCalculus.dir_right_slice_at (hV i)] |
| have hs : directional (EuclideanSpace.basisFun (Fin d) ℝ i) |
| (interpolationPotential F x r ρ) y=jointScore F x r i (ρ,y) := by |
| rw [directional_gradient, |
| interpolationPotential_gradient hF x hr hl hρ0 hρ1] |
| simp only [jointScore,jointMean,inner_add_left,real_inner_smul_left] |
| congr 1 |
| · exact real_inner_comm _ _ |
| · congr 1 |
| exact real_inner_comm _ _ |
| rw [hs] |
| rfl |
|
|
| lemma joint_score_transport {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) |
| (y : Point d) (i : Fin d) : |
| JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointScore F x r i) (ρ,y)= |
| r*JetCalculus.gadj jointSpace (jointScore F x r) |
| (fun k => JetCalculus.dir (jointSpace i) (jointMean F x r k)) (ρ,y) := by |
| have hp : ρ^2<1 := by have := (probability_time hρ0.le hρ1).1; linarith |
| have hm := joint_mdir_eq_material F x r ((jointScore_smooth_at hF x hr.le hl (p:=(ρ,y)) hp i).differentiableAt (by simp)) |
| change JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (jointScore F x r i) (ρ,y)=_ at hm |
| rw [hm] |
| rw [joint_gadj_slice hF x hr.le hl hρ0.le hρ1 (fun k => |
| (JetCalculus.smooth_dir_at (jointMean_smooth_at hF x hr.le hl hp k) _).differentiableAt (by simp))] |
| have he : (fun k z => JetCalculus.dir (jointSpace i) (jointMean F x r k) (ρ,z))= |
| fun k => JetCalculus.dir (EuclideanSpace.basisFun (Fin d) ℝ i) |
| (fun z => inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ k) (conditionalFieldMean F x r ρ z)) := by |
| funext k z |
| exact (JetCalculus.dir_right_slice_at |
| ((jointMean_smooth_at hF x hr.le hl (p:=(ρ,z)) hp k).differentiableAt (by simp)) _).symm |
| rw [he] |
| exact material_interpolation_score hF x hr hlam hl hρ0 hρ1 _ y |
|
|
| theorem physical_material_adjoint {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) |
| {V : Fin d → ℝ × Point d → ℝ} {y : Point d} |
| (hV : ∀i,ContDiffAt ℝ (⊤:ℕ∞) (V i) (ρ,y)) : |
| JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) |
| (JetCalculus.gadj jointSpace (jointScore F x r) V) (ρ,y)= |
| JetCalculus.gadj jointSpace (jointScore F x r) |
| (fun i => JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) (V i)) (ρ,y)+ |
| r*JetCalculus.gadj jointSpace (jointScore F x r) |
| (fun k p => ∑i,JetCalculus.dir (jointSpace i) (jointMean F x r k) p*V i p) (ρ,y) := by |
| have hp : ρ^2<1 := by have := (probability_time hρ0.le hρ1).1; linarith |
| exact JetCalculus.mdir_gadj_at _ _ _ (jointMean_smooth_at hF x hr.le hl hp) |
| (jointScore_smooth_at hF x hr.le hl hp) hV (joint_score_transport hF x hr hlam hl hρ0 hρ1 y) |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
