| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.NodesContinuous |
| import OAI.Probability.LogConcave.Sampling.CenteringJointLaw |
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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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| def probabilityRmsBudget {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (lam A : ℝ≥0) (r h C : ℝ) (k n N : ℕ) : ℝ := |
| 4*C*((lam:ℝ)*r^2)*Real.sqrt d*(1+Real.log ((d:ℝ)+1))^k*(2*h)^(n+1)+ |
| (A*probabilityMeanLipschitz lam r:ℝ≥0)^N*(2*probabilityInitialRms F x lam r) |
|
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| theorem kernel_probability_errors (n : ℕ) (hn : 0<n) : |
| ∃A : ℝ≥0,∃C : ℝ,0≤C ∧ ∃k : ℕ,∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, |
| ∀hF : Primitive F lam,∀(x : Point d) {r : ℝ},∀hr : 0<r, |
| 0<lam → ∀hl : (lam:ℝ)*r^2≤1/2,1≤d → |
| ∀{T h : ℝ},∀hT0 : 0<T,∀hT1 : T<1,∀hh : 0<h,h≤Real.log 2 → |
| A*probabilityMeanLipschitz lam r≤1/2 → ∀N : ℕ,∀s e : ProbabilityNode T h n, |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n e,probabilityNodeTime_mem hT0 hT1 hh hn e⟩ |
| let μ := gibbs (centeringPotential F x r S) |
| let forward := probabilityTransport hF x hr.le hl hT0.le hT1 S E |
| let pos := fun y : Point (d+d) => (productPointEquiv d d y).1 |
| (Integrable (fun y => ‖probabilityAnchoredPicard F x r T h n N e (forward (pos y)) s-pos y‖^2) μ ∧ |
| (∫y,‖probabilityAnchoredPicard F x r T h n N e (forward (pos y)) s-pos y‖^2 ∂μ)≤ |
| (probabilityRmsBudget F x lam A r h C k n N)^2) ∧ |
| ∀Ψ : Point (d+d) → Point (d+d),Measurable Ψ → μ.map Ψ=μ → |
| Integrable (fun y => ‖probabilityAnchoredPicard F x r T h n (N+1) s (pos (Ψ y)) e- |
| forward (pos (Ψ y))‖^2) μ ∧ |
| (∫y,‖probabilityAnchoredPicard F x r T h n (N+1) s (pos (Ψ y)) e- |
| forward (pos (Ψ y))‖^2 ∂μ)≤(probabilityRmsBudget F x lam A r h C k n (N+1))^2 := by |
| obtain ⟨A,C,hC,k,hErr⟩ := probabilityAnchored_marginal_rms n hn |
| refine ⟨A,C,hC,k,?_⟩ |
| intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hq N s e |
| dsimp only |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n e,probabilityNodeTime_mem hT0 hT1 hh hn e⟩ |
| let μ := gibbs (centeringPotential F x r S) |
| let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S E |
| let bw := probabilityTransport hF x hr.le hl hT0.le hT1 E S |
| let pos := fun y : Point (d+d) => (productPointEquiv d d y).1 |
| have hpos : Measurable pos := (productPointEquiv d d).continuous.measurable.fst |
| have hfw : Continuous fw := probabilityTransport_continuous hF x hr.le hl hT0.le hT1 S E |
| have hbw : Continuous bw := probabilityTransport_continuous hF x hr.le hl hT0.le hT1 E S |
| have hp : μ.map pos=interpolationLaw F x r S := |
| centering_position_law hF x hr.le hl S.2.1 (S.2.2.trans_lt hT1) |
| have hfp : μ.map (fun y => fw (pos y))=interpolationLaw F x r E := by |
| change μ.map (fw ∘ pos)=_ |
| rw [←Measure.map_map hfw.measurable hpos,hp] |
| exact probabilityTransport_pushforward hF x hr.le hl hT0.le hT1 S E |
| constructor |
| · have he := hErr hF x hr hlam hl hd hT0 hT1 hh hsmall hq N e s |
| have hm := ((probabilityAnchored_continuous hF x hr.le hl hT0 hT1 hh n hn N e s).sub hbw).measurable |
| have h := rms_bound_pullback (hfw.measurable.comp hpos) hfp hm he.1 he.2 |
| have hc (y : Point d) : bw (fw y)=y := by |
| dsimp [bw,fw] |
| rw [probabilityTransport_cocycle,probabilityTransport_initial] |
| simpa only [Pi.sub_apply,Function.comp_def,hc,probabilityRmsBudget] using h |
| · intro Ψ hm hlaw |
| have he := hErr hF x hr hlam hl hd hT0 hT1 hh hsmall hq (N+1) s e |
| have hΨ := harmonic_position_law hF x hr.le hl S.2.1 (S.2.2.trans_lt hT1) hm hlaw |
| exact rms_bound_pullback (hpos.comp hm) hΨ |
| ((probabilityAnchored_continuous hF x hr.le hl hT0 hT1 hh n hn (N+1) s e).sub hfw).measurable |
| he.1 he.2 |
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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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