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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 | import Mathlib
import OAI.Probability.LogConcave.Analysis.CenteringPotential
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory ProbabilityTheory
open scoped NNReal
lemma centering_joint_law {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) :
(gibbs (centeringPotential F x r T)).map (productPointEquiv d d)=
(interpolationLaw F x r T).prod (stdGaussian (Point d)) := by
rw [interpolationLaw_eq_gibbs hF x hr (by linarith) (by nlinarith)]
exact (productPotential_gibbs
(interpolationPotential_smooth hF x hr hl hT0 hT1).continuous
(gaussianPotential_polySmooth d).smooth.continuous).trans
(by rw [gaussianPotential_gibbs])
lemma centering_position_law {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) :
(gibbs (centeringPotential F x r T)).map (fun y => (productPointEquiv d d y).1)=
interpolationLaw F x r T := by
change (gibbs (centeringPotential F x r T)).map (Prod.fst ∘ (productPointEquiv d d))=_
rw [←Measure.map_map measurable_fst (productPointEquiv d d).continuous.measurable,
centering_joint_law hF x hr hl hT0 hT1]
simp only [Measure.map_fst_prod,measure_univ,one_smul]
lemma harmonic_position_law {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)
{Ψ : Point (d+d) → Point (d+d)} (hm : Measurable Ψ)
(hlaw : (gibbs (centeringPotential F x r T)).map Ψ=gibbs (centeringPotential F x r T)) :
(gibbs (centeringPotential F x r T)).map (fun y => (productPointEquiv d d (Ψ y)).1)=
interpolationLaw F x r T := by
change (gibbs (centeringPotential F x r T)).map ((fun y => (productPointEquiv d d y).1) ∘ Ψ)=_
rw [←Measure.map_map ((productPointEquiv d d).continuous.measurable.fst) hm,hlaw]
exact centering_position_law hF x hr hl hT0 hT1
lemma rms_bound_pullback {X Y E : Type*} [MeasurableSpace X] [MeasurableSpace Y]
[NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
{μ : Measure X} {ν : Measure Y} {f : X → Y} (hf : Measurable f) (hlaw : μ.map f=ν)
{e : Y → E} (he : Measurable e) (hi : Integrable (fun y => ‖e y‖^2) ν)
{B : ℝ} (hB : (∫y,‖e y‖^2 ∂ν)≤B) :
Integrable (fun x => ‖e (f x)‖^2) μ ∧ (∫x,‖e (f x)‖^2 ∂μ)≤B := by
rw [←hlaw] at hi
refine ⟨hi.comp_measurable hf,?_⟩
rw [←integral_map hf.aemeasurable (he.norm.pow_const 2).aestronglyMeasurable,hlaw]
exact hB
end LogConcaveSampling
end
end
end
end OAI
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