openAI-math-dataset / lean /OAI /Probability /LogConcave /Sampling /CenteringVelocityInterpolationRms.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Complexity.CenteringStateBudgetNonneg | |
| import OAI.Probability.LogConcave.Sampling.IdentityOutputEmbedding | |
| import OAI.Probability.LogConcave.Complexity.SingleCellBasisBudget | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Set MeasureTheory Quadrature TensorEnergy | |
| open scoped Classical BigOperators NNReal RealInnerProductSpace | |
| local instance centeringVelocityInterpolationRmsDecidableEqUnit : DecidableEq Unit := Classical.decEq _ | |
| theorem centering_velocity_interpolation_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) (hn : 0<n) {v : ℝ} (hv : v∈Icc (0:ℝ) 1) : | |
| let V := fun t y => skewLieField (centeringPotential F x r T) | |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) (Ξ y t) | |
| Integrable (fun y => ‖V v y-interpolation (probabilityNodes n) | |
| (fun i => V (probabilityNodes n i) y) v‖^2) (gibbs (centeringPotential F x r T)) ∧ | |
| (∫y,‖V v y-interpolation (probabilityNodes n) | |
| (fun i => V (probabilityNodes n i) y) v‖^2 ∂gibbs (centeringPotential F x r T)) ≤ | |
| 4*(singleCellBasisBudget n)^2* | |
| (((d+d:ℕ):ℝ)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)^(n+2)*centeringStateBudget (n+2)) := by | |
| intro V | |
| let A := jointSkew (centeringKernel hF x hr hl hT0 hT1) s | |
| let I := vectorArray (id : Point (d+d) → Point (d+d)) | |
| let J := fun k t y => tensorVector (iterTensorLie (centeringPotential F x r T) A I (k+1)) (Ξ y t) | |
| have hH : PolySmooth (centeringPotential F x r T) := productPotential_polySmooth | |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) | |
| have hA := centeringSkew_polySmooth hF x hr hlam hl hT0 hT1 s | |
| have hI := identityArray_polySmooth (d+d) | |
| have hJ (k : ℕ) (t : ℝ) : Measurable (J k t) := | |
| liePathJet_measurable hH A I hA hI Ξ hm (k+1) t | |
| have hb := singleCell_interpolation_rms n hn (singleCellBasisBudget n) | |
| (singleCellBasisBudget_spec n).1 (singleCellBasisBudget_spec n).2 J hJ | |
| (((d+d:ℕ):ℝ)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)^(n+2)*centeringStateBudget (n+2)) | |
| (by positivity [centeringStateBudget_nonneg (n+2)]) | |
| (fun t ht => by | |
| simpa only [sub_zero,zero_add,Nat.zero_add] using | |
| centering_velocity_taylor_uniform hF x hr hlam hl hR hRT hT0 hT1 hs Ξ hder hm hlaw n 0 t | |
| le_rfl ht.1 ht.2) v hv | |
| let E := fun y => J 0 v y-interpolation (probabilityNodes n) (fun i => J 0 (probabilityNodes n i) y) v | |
| have hE : Measurable E := (hJ 0 v).sub | |
| (Finset.measurable_sum _ (fun i _ => (hJ 0 _).const_smul (basis (probabilityNodes n) i v))) | |
| have he (y : Point (d+d)) : | |
| V v y-interpolation (probabilityNodes n) (fun i => V (probabilityNodes n i) y) v= | |
| identityOutputProjection (d+d) (E y) := by | |
| simp only [E,map_sub,projection_interpolation,J,Nat.zero_add] | |
| simp only [identityOutputProjection_lie _ A hA (jointSkew_skew _ s),V,A,I] | |
| simp_rw [he] | |
| exact projection_rms (identityOutputEmbedding (d+d)) hE hb.1 hb.2 | |
| theorem centering_mean_interpolation_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) (hn : 0<n) {v : ℝ} (hv : v∈Icc (0:ℝ) 1) : | |
| let M := fun t y => conditionalFieldMean F x r T (productPointEquiv d d (Ξ y t)).1 | |
| Integrable (fun y => ‖M v y-interpolation (probabilityNodes n) | |
| (fun i => M (probabilityNodes n i) y) v‖^2) (gibbs (centeringPotential F x r T)) ∧ | |
| (∫y,‖M v y-interpolation (probabilityNodes n) | |
| (fun i => M (probabilityNodes n i) y) v‖^2 ∂gibbs (centeringPotential F x r T)) ≤ | |
| 4*(singleCellBasisBudget n)^2* | |
| ((d*((lam:ℝ)*r)^2)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)^(n+1)*centeringMeanBudget (n+1)) := by | |
| intro M | |
| let A := jointSkew (centeringKernel hF x hr hl hT0 hT1) s | |
| let I := centeringMeanArray F x r T | |
| let J := fun k t y => tensorVector (iterTensorLie (centeringPotential F x r T) A I k) (Ξ y t) | |
| have hH : PolySmooth (centeringPotential F x r T) := productPotential_polySmooth | |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) | |
| have hA := centeringSkew_polySmooth hF x hr hlam hl hT0 hT1 s | |
| have hI := centeringMeanArray_polySmooth hF x hr hlam hl hT0 hT1 | |
| have hJ (k : ℕ) (t : ℝ) : Measurable (J k t) := liePathJet_measurable hH A I hA hI Ξ hm k t | |
| have hb := singleCell_interpolation_rms n hn (singleCellBasisBudget n) | |
| (singleCellBasisBudget_spec n).1 (singleCellBasisBudget_spec n).2 J hJ | |
| ((d*((lam:ℝ)*r)^2)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)^(n+1)*centeringMeanBudget (n+1)) | |
| (by positivity [centeringMeanBudget_nonneg (n+1)]) | |
| (fun t ht => by | |
| simpa only [J,iterTensorLie,sub_zero] using | |
| centering_mean_taylor_uniform hF x hr hlam hl hR hRT hT0 hT1 hs Ξ hder hm hlaw n 0 t | |
| le_rfl ht.1 ht.2) v hv | |
| let E := fun y => J 0 v y-interpolation (probabilityNodes n) (fun i => J 0 (probabilityNodes n i) y) v | |
| have hE : Measurable E := (hJ 0 v).sub | |
| (Finset.measurable_sum _ (fun i _ => (hJ 0 _).const_smul (basis (probabilityNodes n) i v))) | |
| have he (y : Point (d+d)) : | |
| M v y-interpolation (probabilityNodes n) (fun i => M (probabilityNodes n i) y) v= | |
| meanOutputProjection d (E y) := by | |
| simp only [E,map_sub,projection_interpolation,J,iterTensorLie,I,meanOutputProjection_mean,M] | |
| simp_rw [he] | |
| exact projection_rms (meanOutputEmbedding d) hE hb.1 hb.2 | |
| end LogConcaveSampling | |
| end | |
| end | |
| end OAI | |