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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 | import Mathlib
import OAI.Probability.LogConcave.OraclePrograms.FullTape
import OAI.Probability.LogConcave.Sampling.ProdMapEventReal
namespace OAI
section
noncomputable section
namespace LogConcaveSampling.OracleCompiler
open MeasureTheory ProbabilityTheory Coupling Function
open scoped Classical NNReal
variable {d : ℕ}
lemma ReservedProgram.tv_of_pre {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam)
(S : ReservedProgram d) (x : Point d) {r η e : ℝ}
(hr : 0≤r) (hl : (lam:ℝ)*r^2<1) (hη : 0<η) (he : 0≤e)
(hres : S.reserve=η/2)
(hpre : SquaredAt (gaussianTape d S.slots)
((gibbs (primitivePotential F x r)).prod (stdGaussian (Point d)))
(fun g => S.pre.run F (x,g)) (fun z => z.1+(Real.sqrt 3*η/2) • z.2)
(e^2*(circuitD F x)^2)) :
TVAtMost ((gaussianTape d S.full.slots).map (fun g => S.full.program.run F (x,g)))
(((gibbs (primitivePotential F x r)).prod (stdGaussian (Point d))).map
(fun z => z.1+η • z.2)) (2*e*circuitD F x/η) := by
let := probability_gibbs_of_partition
(partition_pos_of_continuous (hF.continuous_potential x r)).ne'
(partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr hl))
have hV : Measurable (fun z => (F z,gradient F z)) :=
hF.smooth.continuous.measurable.prodMk hF.gradient_lipschitz.continuous.measurable
have hf : Measurable (fun g => S.pre.run F (x,g)) :=
(S.pre.measurable_run F hV).comp (by fun_prop)
have hh := hpre.gaussian_smoothing hf (by fun_prop) (show 0<η/2 by positivity)
have hfull : ((gaussianTape d S.slots).prod (stdGaussian (Point d))).map
(fun z => S.pre.run F (x,z.1)+(η/2) • z.2)=
(gaussianTape d S.full.slots).map (fun g => S.full.program.run F (x,g)) := by
let f : (Fin (S.slots+1) → Point d) → Point d := fun g => S.full.program.run F (x,g)
have hfm : Measurable f :=
(S.full.program.measurable_run F hV).comp (measurable_const.prodMk measurable_id)
have hp := ReservedProgram.fullTape_preserving (d:=d) S.slots
have heq : (fun z : (Fin S.slots → Point d) × Point d =>
S.pre.run F (x,z.1)+(η/2) • z.2)=f ∘ ReservedProgram.fullTape S.slots := by
funext z
dsimp only [f,comp_apply]
rw [ReservedProgram.full_run]
simp only [ReservedProgram.fullTape,Fin.append_left,Fin.append_right,hres]
exact (congrArg (fun k => ((gaussianTape d S.slots).prod (stdGaussian (Point d))).map k) heq).trans
((Measure.map_map hfm hp.measurable).symm.trans (congrArg (Measure.map f) hp.map_eq))
have hnoise := smoothing_add_gaussian
((gibbs (primitivePotential F x r)).prod (stdGaussian (Point d))) (gibbs (primitivePotential F x r))
(fun z => z.1+(Real.sqrt 3*η/2) • z.2) id (by fun_prop) measurable_id
(Real.sqrt 3*η/2) (η/2) η (by
rw [div_pow,mul_pow,Real.sq_sqrt (by norm_num : (0:ℝ)≤3),div_pow]; ring) rfl
rw [hfull,hnoise] at hh
have hs : Real.sqrt (e^2*(circuitD F x)^2)=e*circuitD F x := by
rw [←mul_pow,Real.sqrt_sq (mul_nonneg he (circuitD_nonneg F x))]
have heq : e*circuitD F x/(η/2)=2*e*circuitD F x/η := by ring
simpa only [id_eq,hs,heq] using hh
end LogConcaveSampling.OracleCompiler
end
end
end OAI
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