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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | 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
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