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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 | import Mathlib
import OAI.Probability.LogConcave.Numerics.CompositeAngle
import OAI.Probability.LogConcave.Complexity.HarmonicCorrectionBudget
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 _
def angleNodes (n : ℕ) (i : Fin (n+1)) : ℝ := (i:ℝ)
lemma angleNodes_injective (n : ℕ) : Function.Injective (angleNodes n) := by
intro i j h
apply Fin.ext
change (i:ℝ)=(j:ℝ) at h
exact_mod_cast h
lemma liePathJet_measurable {S : Type} [Fintype S] {d : ℕ} {H : Point d → ℝ}
(hH : PolySmooth H) (A : (Unit ⊕ Unit → Fin d) → Point d → ℝ)
(F : (S → Fin d) → Point d → ℝ) (hA : ∀c,PolySmooth (A c)) (hF : ∀c,PolySmooth (F c))
(Ξ : Point d → ℝ → Point d) (hm : Measurable (fun p : ℝ × Point d => Ξ p.2 p.1))
(k : ℕ) (v : ℝ) :
Measurable (fun y => tensorVector (iterTensorLie H A F k) (Ξ y v)) :=
(tensorVector_contDiff _ (fun c => (iterTensorLie_polySmooth hH A F hA hF k c).smooth)).continuous.measurable.comp
(hm.comp (measurable_const.prodMk measurable_id))
lemma chainTaylor_measurable {Ω E : Type*} [MeasurableSpace Ω]
[NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E]
[SecondCountableTopology E] (J : ℕ → ℝ → Ω → E)
(hm : ∀k t,Measurable (J k t)) (n : ℕ) (a b : ℝ) :
Measurable (fun y => chainTaylor (fun k t => J k t y) n a b) := by
unfold chainTaylor
exact Finset.measurable_sum _ (fun k _ => (hm k a).const_smul _)
theorem harmonic_correction_stencil_rms {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r R T : 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)
(s t u : Icc (0:ℝ) T)
(Ξ : Point (d+d) → ℝ → Point (d+d))
(hder : ∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y)
(skewLieField (centeringPotential F x r u) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v)
(hm : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1))
(hlaw : ∀v∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r u)).map (fun y => Ξ y v)=
gibbs (centeringPotential F x r u)) :
let A := jointCorrectionArray (fun z => probabilityTransport hF x hr.le hl hT0 hT1 s t z-z)
let J := fun k v y => tensorVector (iterTensorLie (centeringPotential F x r u) (harmonicSkew d) A k) (Ξ y v)
∀n : ℕ,1≤n → ∀ψ : ℝ,0<ψ → ψ*n≤1 →
Integrable (fun y => ‖derivativeStencil (angleNodes n) ψ (fun i => J 0 (ψ*angleNodes n i) y)-J 1 0 y‖^2)
(gibbs (centeringPotential F x r u)) ∧
(∫y,‖derivativeStencil (angleNodes n) ψ (fun i => J 0 (ψ*angleNodes n i) y)-J 1 0 y‖^2
∂gibbs (centeringPotential F x r u))≤
((∑i,|derivativeWeight (angleNodes n) i|)/ψ)^2*
(((ψ*n)^(n+1)/(n.factorial:ℝ))^2*
((d*((lam:ℝ)*r^2)^2)*(R⁻¹)^(4*(n+1))*harmonicCorrectionBudget (n+1))) := by
intro A J n hn ψ hψ hψn
have hu1 := u.2.2.trans_lt hT1
have hH : PolySmooth (centeringPotential F x r u) := productPotential_polySmooth
(interpolationPotential_polySmooth hF x hr hlam hl u.2.1 hu1) (gaussianPotential_polySmooth d)
have hJ (k : ℕ) (v : ℝ) : Measurable (J k v) :=
liePathJet_measurable hH (harmonicSkew d) A (harmonicSkew_polySmooth d)
(jointCorrectionArray_polySmooth hF x hr hlam hl hR hRT hT0 hT1 s t) Ξ hm k v
have hnode (i : Fin (n+1)) : 0≤ψ*angleNodes n i ∧ ψ*angleNodes n i≤ψ*n := by
constructor
· exact mul_nonneg hψ.le (Nat.cast_nonneg _)
· apply mul_le_mul_of_nonneg_left _ hψ.le
change (i:ℝ)≤(n:ℝ)
exact_mod_cast Nat.le_of_lt_succ i.2
have htay (i : Fin (n+1)) := harmonic_correction_taylor_rms hF x hr hlam hl hR hRT hT0 hT1
s t u Ξ hder hm hlaw n 0 (ψ*angleNodes n i) le_rfl (hnode i).1 ((hnode i).2.trans hψn)
apply derivative_stencil_error_rms (angleNodes n) (angleNodes_injective n) n hn (by simp) hψ J
· intro i
exact ((hJ 0 _).sub (chainTaylor_measurable J hJ n 0 _)).aestronglyMeasurable
· intro i
simpa only [J,iterTensorLie,sub_zero] using (htay i).1
· intro i
have hh := (htay i).2
change (∫y,‖J 0 (ψ*angleNodes n i) y-chainTaylor
(fun k v => J k v y) n 0 (ψ*angleNodes n i)‖^2 ∂gibbs (centeringPotential F x r u))≤_ at hh
apply hh.trans
simp only [sub_zero]
apply mul_le_mul_of_nonneg_right _ (by positivity [harmonicCorrectionBudget_nonneg (n+1)])
apply pow_le_pow_left₀ (div_nonneg (pow_nonneg (hnode i).1 _) (Nat.cast_nonneg _))
apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
exact pow_le_pow_left₀ (hnode i).1 (hnode i).2 _
end LogConcaveSampling
end
end
end
end OAI
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