| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.FlowRestrict |
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|
| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped NNReal RealInnerProductSpace |
|
|
| theorem centering_output_law {d : ℕ} {μ : Measure (Point d)} [IsProbabilityMeasure μ] |
| {K : Point d → Point d →L[ℝ] Point d} {M : Point d → Point d} |
| (hM : Continuous M) (u : Point d) {s : ℝ} (hs : 0<s) |
| {Ψ : (Point d × Point d) → ℝ → (Point d × Point d)} |
| (hΨ : ∀p,Continuous (Ψ p) ∧ Ψ p 0=p ∧ ∀t∈Icc (0:ℝ) 1, |
| HasDerivWithinAt (Ψ p) (skewCenteringField K (fun y => M y-u) s (Ψ p t)) (Icc (0:ℝ) 1) t) |
| (hΨc : Continuous (fun p => Ψ p 1)) |
| (hΨlaw : (μ.prod (stdGaussian (Point d))).map (fun p => Ψ p 1)=μ.prod (stdGaussian (Point d))) : |
| (μ.prod (stdGaussian (Point d))).map |
| (fun p => s • p.2+∫t in (0:ℝ)..1,M (Ψ p t).1)= |
| (stdGaussian (Point d)).map (fun g => u+s • g) := by |
| have he : (fun p => s • p.2+∫t in (0:ℝ)..1,M (Ψ p t).1)= |
| (fun p => u+s • (Ψ p 1).2) := by |
| funext p |
| simpa only [(hΨ p).2.1] using centering_momentum_identity hM u hs (hΨ p).1 (hΨ p).2.2 |
| rw [he] |
| have hm : Measurable (fun p : Point d × Point d => u+s • p.2) := by fun_prop |
| change (μ.prod (stdGaussian (Point d))).map |
| ((fun p : Point d × Point d => u+s • p.2) ∘ (fun p => Ψ p 1)) = _ |
| rw [←Measure.map_map hm hΨc.measurable,hΨlaw] |
| rw [show (fun p : Point d × Point d => u+s • p.2)= |
| (fun g : Point d => u+s • g) ∘ Prod.snd from rfl, |
| ←Measure.map_map (by fun_prop) measurable_snd,Measure.map_snd_prod, |
| measure_univ,one_smul] |
| 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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