| import Mathlib |
| import OAI.Probability.LogConcave.Analysis.CenteringPotential |
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|
| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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|
| 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]) |
|
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| 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 |
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| 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 |
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| end LogConcaveSampling |
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| end |
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| end |
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| end |
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| end OAI |
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