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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 | import Mathlib
import OAI.Probability.LogConcave.Dynamics.FlowRestrict
namespace OAI
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
end
end
end
end OAI
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