| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.CenteringStateQuadratureBudget |
| import OAI.Probability.LogConcave.Dynamics.ObservableIntegralMeasurable |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory Quadrature RMSIntegral |
| open scoped Classical BigOperators NNReal |
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| def centeringMeanCircuit {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (r T h ψ s : ℝ) (n m N nc Nc : ℕ) (e : ProbabilityNode T h (n+1)) |
| (y : Point (d+d)) : Point d := |
| s • (productPointEquiv d d y).2+ |
| ∑j,weight (probabilityNodes nc) j 0 1 • conditionalFieldMean F x r T |
| (productPointEquiv d d (stationaryPicard nc Nc |
| (centeringVelocityJoint F x r T h ψ s n m N e) y j)).1 |
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| def centeringInitialBudget (d : ℕ) (lam : ℝ≥0) (r R s : ℝ) : ℝ := |
| (((d+d:ℕ):ℝ)*(((lam:ℝ)*r/s)^2*(R⁻¹)^4)*centeringStateBudget 1) |
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| def centeringPicardBudget (nc Nc d : ℕ) (lam K : ℝ≥0) (r R s Bv : ℝ) : ℝ := |
| (2*Real.sqrt (2*(centeringRowBudget nc:ℝ)^2*Bv+ |
| 2*centeringStateQuadratureBudget nc d lam r R s)+ |
| ((centeringRowBudget nc*K:ℝ≥0):ℝ)^Nc*Real.sqrt (centeringInitialBudget d lam r R s))^2 |
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| variable {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r R T h ψ s : ℝ} |
| (hr : 0<r) (hlam : 0<lam) (hl : (lam:ℝ)*r^2 ≤ 1/2) |
| (hR : 0<R) (hRT : R^2 ≤ 1-T^2) (hT0 : 0 ≤ T) (hT1 : T<1) (hs : 0<s) |
| (X : StationaryPath (gibbs (centeringPotential F x r T)) |
| (skewLieField (centeringPotential F x r T) (jointSkew (centeringKernel hF x hr hl hT0 hT1) s))) |
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| include hlam hR hRT hs |
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| theorem centeringPicard_state_rms (n m N nc Nc : ℕ) (hnc : 0<nc) |
| (e : ProbabilityNode T h (n+1)) {K : ℝ≥0} |
| (hhat : LipschitzWith K (centeringVelocityJoint F x r T h ψ s n m N e)) |
| (hq : centeringRowBudget nc*K ≤ 1/2) {Bv : ℝ} (hBv : 0 ≤ Bv) |
| (hvi : Integrable (fun y => ‖centeringVelocityJoint F x r T h ψ s n m N e y- |
| skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) y‖^2) |
| (gibbs (centeringPotential F x r T))) |
| (hvb : (∫y,‖centeringVelocityJoint F x r T h ψ s n m N e y- |
| skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) y‖^2 |
| ∂gibbs (centeringPotential F x r T)) ≤ Bv) (i : Fin (nc+1)) : |
| Integrable (fun y => ‖stationaryPicard nc Nc |
| (centeringVelocityJoint F x r T h ψ s n m N e) y i-X.path y (probabilityNodes nc i)‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖stationaryPicard nc Nc (centeringVelocityJoint F x r T h ψ s n m N e) y i- |
| X.path y (probabilityNodes nc i)‖^2 ∂gibbs (centeringPotential F x r T)) ≤ |
| centeringPicardBudget nc Nc d lam K r R s Bv := by |
| have hH := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) |
| have ht := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) (gaussianPotential_lowerTail d) |
| let : IsProbabilityMeasure (gibbs (centeringPotential F x r T)) := |
| probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| exact stationaryPicard_rms X (centeringTrueVelocity_continuous hF x hr hlam hl hT0 hT1 s) |
| hhat (centering_lipschitz_memLp hF x hr hlam hl hT0 hT1 (LipschitzWith.id)) nc hnc hq |
| (centeringStateQuadratureBudget_nonneg nc d lam r R s) hBv |
| (by positivity [centeringStateBudget_nonneg 1]) |
| (centering_state_quadrature_rms hF x hr hlam hl hR hRT hT0 hT1 hs X nc hnc) hvi hvb |
| (fun j => (centering_initial_rms hF x hr hlam hl hR hRT hT0 hT1 hs X |
| (probabilityNodes_mem hnc j)).2) Nc i |
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| theorem centeringMeanCircuit_rms (n m N nc Nc : ℕ) (hnc : 0<nc) |
| (e : ProbabilityNode T h (n+1)) {K : ℝ≥0} |
| (hhat : LipschitzWith K (centeringVelocityJoint F x r T h ψ s n m N e)) |
| {Bz : ℝ} (hBz : 0 ≤ Bz) |
| (hZi : ∀j, Integrable (fun y => ‖stationaryPicard nc Nc |
| (centeringVelocityJoint F x r T h ψ s n m N e) y j-X.path y (probabilityNodes nc j)‖^2) |
| (gibbs (centeringPotential F x r T))) |
| (hZb : ∀j, (∫y,‖stationaryPicard nc Nc |
| (centeringVelocityJoint F x r T h ψ s n m N e) y j-X.path y (probabilityNodes nc j)‖^2 |
| ∂gibbs (centeringPotential F x r T)) ≤ Bz) : |
| let ideal := fun y => s • (productPointEquiv d d y).2+ |
| ∫t in (0:ℝ)..1,conditionalFieldMean F x r T (productPointEquiv d d (X.path y t)).1 |
| Integrable (fun y => ‖centeringMeanCircuit F x r T h ψ s n m N nc Nc e y-ideal y‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖centeringMeanCircuit F x r T h ψ s n m N nc Nc e y-ideal y‖^2 |
| ∂gibbs (centeringPotential F x r T)) ≤ |
| 2*(centeringRowBudget nc:ℝ)^2*((Real.pi^2/2)*T*((lam:ℝ)*r))^2*Bz+ |
| 2*centeringMeanQuadratureBudget nc d lam r R s := by |
| intro ideal |
| have hH := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) |
| have ht := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) (gaussianPotential_lowerTail d) |
| let : IsProbabilityMeasure (gibbs (centeringPotential F x r T)) := |
| probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| have hMc : LipschitzWith ⟨(Real.pi^2/2)*T*((lam:ℝ)*r),by positivity⟩ |
| (fun y => conditionalFieldMean F x r T (productPointEquiv d d y).1) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro y z |
| have hp : dist (productPointEquiv d d y).1 (productPointEquiv d d z).1 ≤ dist y z := by |
| simpa only [dist_eq_norm,map_sub,Prod.fst_sub] using productPointEquiv_fst_norm_le (y-z) |
| exact ((conditionalFieldMean_lipschitz hF x hr.le hl hT0 hT1).dist_le_mul _ _).trans |
| (mul_le_mul_of_nonneg_left hp (NNReal.coe_nonneg _)) |
| have hquad := centering_mean_quadrature_rms hF x hr hlam hl hR hRT hT0 hT1 hs X nc hnc |
| have he := stationaryOutput_rms X hMc nc |
| (fun j y => stationaryPicard nc Nc (centeringVelocityJoint F x r T h ψ s n m N e) y j) |
| (stationaryPicard_measurable nc Nc hhat.continuous.measurable) hBz hZi hZb hquad.1 hquad.2 |
| (fun y => s • (productPointEquiv d d y).2) |
| exact he |
| end LogConcaveSampling |
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| end |
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| end |
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| end OAI |
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