| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.ConditionalMeanArray |
| import OAI.Probability.LogConcave.Dynamics.ActualJointCenteringFlow |
| import OAI.Probability.LogConcave.Sampling.StationaryTensorVectorRms |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
| open TensorEnergy Quadrature |
|
|
| local instance : DecidableEq Unit := Classical.decEq _ |
|
|
| lemma gradient_gaussianPotential {d : ℕ} (y : Point d) : |
| gradient (fun z : Point d => ‖z‖^2/2) y=y := by |
| ext i |
| rw [←directional_basis_gradient,directional_gaussianPotential] |
| simp [EuclideanSpace.basisFun_apply,PiLp.inner_apply] |
|
|
| lemma gaussianPotential_gradient_lipschitz (d : ℕ) : |
| LipschitzWith 1 (gradient (fun z : Point d => ‖z‖^2/2)) := by |
| have he : gradient (fun z : Point d => ‖z‖^2/2)=id := funext gradient_gaussianPotential |
| rw [he] |
| exact LipschitzWith.id |
|
|
| def centeringPotential {d : ℕ} (F : Point d → ℝ) (x : Point d) (r T : ℝ) : Point (d+d) → ℝ := |
| productPotential (interpolationPotential F x r T) (fun z : Point d => ‖z‖^2/2) |
|
|
| def centeringMeanArray {d : ℕ} (F : Point d → ℝ) (x : Point d) (r T : ℝ) : |
| (Unit → Fin (d+d)) → Point (d+d) → ℝ := |
| arrayCoordinateLift (leftCoordinates d d) (fun _ => leftCoordinates d d) |
| (conditionalMeanArray F x r T) |
|
|
| def centeringGradientBound (lam : ℝ≥0) (r T : ℝ) (h : 0≤(Real.pi^2/2)*T*((lam:ℝ)*r)) : ℝ≥0 := |
| (1+‖r*T‖₊*⟨(Real.pi^2/2)*T*((lam:ℝ)*r),h⟩)+1 |
|
|
| lemma centeringPotential_gradient_lipschitz {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) : |
| LipschitzWith (centeringGradientBound lam r T (by positivity)) |
| (gradient (centeringPotential F x r T)) := |
| productPotential_gradient_lipschitz |
| ((interpolationPotential_smooth hF x hr.le hl hT0 hT1).differentiable (by simp)) |
| ((gaussianPotential_polySmooth d).smooth.differentiable (by simp)) |
| (interpolationPotential_gradient_lipschitz hF x hr.le hl hT0 hT1) |
| (gaussianPotential_gradient_lipschitz d) |
|
|
| lemma centeringMeanArray_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 → Fin (d+d)) : PolySmooth (centeringMeanArray F x r T c) := |
| arrayCoordinateLift_polySmooth _ _ _ (conditionalMeanArray_polySmooth hF x hr hlam hl hT0 hT1) c |
|
|
| lemma centeringMeanArray_energy {d n : 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) |
| (hn : 0<n) : |
| spatialEnergy (centeringMeanArray F x r T) n (gibbs (centeringPotential F x r T))≤ |
| d*((lam:ℝ)*r)^2*(normalizedTensorMajorant (n+1) (1-T^2))^2 := by |
| have hH := interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1 |
| have ht := interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1 |
| let := probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| dsimp only [centeringMeanArray,centeringPotential] |
| rw [spatialEnergy_leftLift hH.smooth.continuous _ |
| (fun c => (conditionalMeanArray_polySmooth hF x hr hlam hl hT0 hT1 c).smooth)] |
| exact conditionalMeanArray_energy hF x hr hlam hl hT0 hT1 hn |
|
|
| lemma centeringSkew_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) (s : ℝ) : |
| ∀c,PolySmooth (jointSkew (centeringKernel hF x hr hl hT0 hT1) s c) := by |
| apply jointSkew_polySmooth |
| intro i j |
| simpa only [matrixArray_apply,Sum.elim_inl,Sum.elim_inr] using |
| centeringKernel_polySmooth hF x hr hlam hl hT0 hT1 (Sum.elim (fun _ => i) (fun _ => j)) |
|
|
| lemma centeringSkew_scaled_bound {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T s : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hR : 0<R) (hRT : R^2≤1-T^2) |
| (hT0 : 0≤T) (hT1 : T<1) (hs : 0<s) : |
| ∀k y,AllSplitBound |
| (spatialTensor (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (List.finRange k) y) |
| ((2*kernelMajorant k/R^k)*((lam:ℝ)*r/s)*1^k) := by |
| intro k y |
| convert centering_jointSkew_spatial_bound hF x hr hlam hl hR hRT hT0 hT1 s y using 1 |
| rw [abs_of_pos (inv_pos.mpr hs)] |
| simp only [one_pow,mul_one] |
| ring |
|
|
| lemma centeringScore_scaled_bound {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) : |
| ∀k,0<k → ∀y,AllSplitBound |
| (spatialTensor (scoreField (centeringPotential F x r T)) (List.finRange k) y) |
| ((2+normalizedTensorMajorant (k+1) (1-T^2))*1^k) := by |
| intro k hk y |
| simpa only [centeringPotential,one_pow,mul_one] using |
| centeringScore_allSplit hF x hr hlam hl hT0 hT1 hk y |
|
|
| lemma centeringMean_scaled_energy {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) : |
| ∀k,0<k → spatialEnergy (centeringMeanArray F x r T) k |
| (gibbs (centeringPotential F x r T))≤ |
| (d*((lam:ℝ)*r)^2)*1^(2*k)*(normalizedTensorMajorant (k+1) (1-T^2))^2 := by |
| intro k hk |
| simpa only [one_pow,mul_one] using centeringMeanArray_energy hF x hr hlam hl hT0 hT1 hk |
|
|
| theorem centering_mean_taylor_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T s : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hR : 0<R) (hRT : R^2≤1-T^2) |
| (hT0 : 0≤T) (hT1 : T<1) (hs : 0<s) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) |
| (hder : ∀y t,t∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (Ξ y t)) (Icc (0:ℝ) 1) t) |
| (hm : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1)) |
| (hlaw : ∀t∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r T)).map (fun y => Ξ y t)= |
| gibbs (centeringPotential F x r T)) : |
| ∀n : ℕ,∀a b : ℝ,0≤a → a≤b → b≤1 → |
| Integrable (fun y => ‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2 |
| ∂gibbs (centeringPotential F x r T))≤ |
| ((b-a)^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(((lam:ℝ)*r/s)^2)^(n+1)* |
| iterEnergyBudget (centeringGradientBound lam r T (by positivity)) |
| (fun k => 2*kernelMajorant k/R^k) |
| (fun k => 2+normalizedTensorMajorant (k+1) (1-T^2)) |
| (fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2) (n+1) 0) := by |
| have hH : PolySmooth (centeringPotential F x r T) := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) |
| have ht : HasGaussianLowerTail (centeringPotential F x r T) := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) (gaussianPotential_lowerTail d) |
| let := probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| intro n a b ha hab hb |
| have hsub : uIcc a b⊆Icc (0:ℝ) 1 := by |
| rw [uIcc_of_le hab] |
| intro t h; exact ⟨ha.trans h.1,h.2.trans hb⟩ |
| have h := stationary_lie_taylor_rms (S:=Unit) (d:=d+d) |
| (H:=centeringPotential F x r T) hH ht |
| (centeringPotential_gradient_lipschitz hF x hr hl hT0 hT1) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (centeringMeanArray F x r T) |
| (centeringSkew_polySmooth hF x hr hlam hl hT0 hT1 s) (centeringMeanArray_polySmooth hF x hr hlam hl hT0 hT1) |
| (jointSkew_skew _ s) |
| (fun k => 2*kernelMajorant k/R^k) |
| (fun k => 2+normalizedTensorMajorant (k+1) (1-T^2)) |
| (fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2) |
| (α:=(lam:ℝ)*r/s) (κ:=d*((lam:ℝ)*r)^2) (ρ:=1) (by positivity) le_rfl |
| (fun _ => sq_nonneg _) |
| (centeringSkew_scaled_bound hF x hr hlam hl hR hRT hT0 hT1 hs) |
| (centeringScore_scaled_bound hF x hr hlam hl hT0 hT1) |
| (centeringMean_scaled_energy hF x hr hlam hl hT0 hT1) |
| Ξ hder hm hlaw hab hsub n |
| refine ⟨h.1,h.2.trans ?_⟩ |
| simp only [one_pow,mul_one] |
| exact le_rfl |
|
|
| theorem exists_centering_mean_taylor_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T s : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hR : 0<R) (hRT : R^2≤1-T^2) |
| (hT0 : 0≤T) (hT1 : T<1) (hs : 0<s) : |
| ∃Ξ : Point (d+d) → ℝ → Point (d+d), |
| (∀y,Continuous (Ξ y) ∧ Ξ y 0=y) ∧ |
| Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1) ∧ |
| (∀t∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r T)).map (fun y => Ξ y t)= |
| gibbs (centeringPotential F x r T)) ∧ |
| ∀n : ℕ,∀a b : ℝ,0≤a → a≤b → b≤1 → |
| Integrable (fun y => ‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖tensorVector (centeringMeanArray F x r T) (Ξ y b)-chainTaylor |
| (fun k t => tensorVector (iterTensorLie (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) |
| (centeringMeanArray F x r T) k) (Ξ y t)) n a b‖^2 |
| ∂gibbs (centeringPotential F x r T))≤ |
| ((b-a)^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(((lam:ℝ)*r/s)^2)^(n+1)* |
| iterEnergyBudget (centeringGradientBound lam r T (by positivity)) |
| (fun k => 2*kernelMajorant k/R^k) |
| (fun k => 2+normalizedTensorMajorant (k+1) (1-T^2)) |
| (fun k => (normalizedTensorMajorant (k+1) (1-T^2))^2) (n+1) 0) := by |
| obtain ⟨Ξ,hinit,hder,hm,hlaw⟩ := exists_actual_joint_centering_flow hF x hr hlam hl hT0 hT1 hs |
| exact ⟨Ξ,hinit,hm,hlaw,centering_mean_taylor_rms hF x hr hlam hl hR hRT hT0 hT1 hs Ξ hder hm hlaw⟩ |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
