| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.KernelActionPicardRms |
| import OAI.Probability.LogConcave.Sampling.ProbabilityEndpoint |
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| namespace OAI |
|
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory RMSIntegral Quadrature |
| open scoped Classical BigOperators NNReal |
|
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| lemma kernelActionPicard_continuous {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T h : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (ψ : ℝ) (n : ℕ) (hn : 0<n) (m N : ℕ) (s e : ProbabilityNode T h n) : |
| Continuous (kernelActionPicard F x r T h ψ n m N s e) := by |
| change Continuous (fun p => ∑i,(derivativeWeight (angleNodes m) i/ψ) • |
| kernelCorrectionPicard F x r T h (ψ*angleNodes m i) n N s e p) |
| apply continuous_finsetSum |
| intro i _ |
| have hc : Continuous (fun v : Point d => (derivativeWeight (angleNodes m) i/ψ) • v) := continuous_const_smul _ |
| exact hc.comp (kernelCorrectionPicard_continuous hF x hr hl hT0 hT1 hh (ψ*angleNodes m i) n hn N s e) |
|
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| def kernelActionRmsBudget {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (lam Ap Ah Lp : ℝ≥0) (r R ρ h ψ C J : ℝ) (k n m N : ℕ) : ℝ := |
| 2*((∑i,|derivativeWeight (angleNodes m) i/ψ|)* |
| (∑i,|derivativeWeight (angleNodes m) i/ψ| * |
| kernelValueRmsBudget F x lam Ap Ah Lp r R ρ h (ψ*angleNodes m i) C J k n N))+ |
| 2*(((∑i,|derivativeWeight (angleNodes m) i|)/ψ)^2* |
| (((ψ*m)^(m+1)/(m.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r^2)^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)))) |
|
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| theorem kernelActionPicard_terminal_rms (n : ℕ) (hn : 0<n) : |
| ∃Ap Ah Lp Lh : ℝ≥0,∃C J : ℝ,0≤C ∧ 1≤J ∧ ∃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 → |
| Ap*probabilityMeanLipschitz lam r≤1/2 → Ah*probabilityMeanLipschitz lam r≤1/2 → |
| Lp*probabilityMeanLipschitz lam r≤1/2 → Lh*probabilityMeanLipschitz lam r≤1/2 → |
| ∀N : ℕ,∀s : ProbabilityNode T h n, |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| ∀R : ℝ,0<R → R^2≤1-T^2 → ∀m : ℕ,1 ≤ m → ∀ψ : ℝ,0<ψ → ψ*m≤1 → |
| let err := fun p : Point d × Point d => |
| kernelActionPicard F x r T h ψ n m N s (probabilityEndpoint hT0 hT1 hh n) p- |
| (terminalJacobian hF x hr hl hT0.le hT1 (S,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2 |
| Integrable (fun p => ‖err p‖^2) ((interpolationLaw F x r T).prod (stdGaussian (Point d))) ∧ |
| (∫p,‖err p‖^2 ∂((interpolationLaw F x r T).prod (stdGaussian (Point d))))≤ |
| kernelActionRmsBudget F x lam Ap Ah Lp r R S h ψ C J k n m N := by |
| obtain ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,hA⟩ := kernelActionPicard_rms n hn |
| refine ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,?_⟩ |
| intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s |
| dsimp only |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let ep := probabilityEndpoint hT0 hT1 hh n |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n ep,probabilityNodeTime_mem hT0 hT1 hh hn ep⟩ |
| have hE : E=⟨T,hT0.le,le_rfl⟩ := Subtype.ext (probabilityEndpoint_time hT0 hT1 hh n hn) |
| intro R hR hRT m hm1 ψ hψ hψm |
| let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S ⟨T,hT0.le,le_rfl⟩ |
| let inp := fun y : Point (d+d) => (fw (productPointEquiv d d y).1,(productPointEquiv d d y).2) |
| let err := fun p : Point d × Point d => kernelActionPicard F x r T h ψ n m N s ep p- |
| (terminalJacobian hF x hr hl hT0.le hT1 (S,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2 |
| have ha := hA hF x hr hlam hl hd hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s ep R hR hRT m hm1 ψ hψ hψm |
| let aerr := fun f : Point d → Point d => fun y : Point (d+d) => |
| kernelActionPicard F x r T h ψ n m N s ep |
| (f (productPointEquiv d d y).1,(productPointEquiv d d y).2)-kernelTransportAction f y |
| change Integrable (fun y => ‖aerr (probabilityTransport hF x hr.le hl hT0.le hT1 S E) y‖^2) |
| (gibbs (centeringPotential F x r S)) ∧ |
| (∫y,‖aerr (probabilityTransport hF x hr.le hl hT0.le hT1 S E) y‖^2 |
| ∂gibbs (centeringPotential F x r S))≤kernelActionRmsBudget F x lam Ap Ah Lp r R S h ψ C J k n m N at ha |
| rw [hE] at ha |
| have heq (y : Point (d+d)) : err (inp y)=kernelActionPicard F x r T h ψ n m N s ep (inp y)-kernelTransportAction fw y := by |
| dsimp only [err,inp] |
| rw [kernelTransportAction_terminal,terminalForward_eq] |
| have hfw : Measurable fw := (probabilityTransport_continuous hF x hr.le hl hT0.le hT1 S ⟨T,hT0.le,le_rfl⟩).measurable |
| have hinp : Measurable inp := (hfw.comp ((productPointEquiv d d).continuous.measurable.fst)).prodMk |
| ((productPointEquiv d d).continuous.measurable.snd) |
| have hlaw := centering_transport_joint_law hF x hr.le hl hT0.le hT1 S ⟨T,hT0.le,le_rfl⟩ |
| have hK : Continuous (fun p : Point d × Point d => |
| (terminalJacobian hF x hr hl hT0.le hT1 (S,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2) := |
| (((terminalJacobian_smooth hF x hr hl hT0.le hT1).continuous.comp |
| (continuous_const.prodMk continuous_fst)).sub continuous_const).clm_apply continuous_snd |
| have he : Measurable err := ((kernelActionPicard_continuous hF x hr.le hl hT0 hT1 hh ψ n hn m N s ep).sub hK).measurable |
| have hi : Integrable (fun y => ‖err (inp y)‖^2) (gibbs (centeringPotential F x r S)) := by |
| simpa only [heq] using ha.1 |
| have hb : (∫y,‖err (inp y)‖^2 ∂gibbs (centeringPotential F x r S))≤ |
| kernelActionRmsBudget F x lam Ap Ah Lp r R S h ψ C J k n m N := by |
| simpa only [heq] using ha.2 |
| exact rms_bound_pushforward hinp hlaw he hi hb |
| end LogConcaveSampling |
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|
| end |
|
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| end |
|
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| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory RMSIntegral Quadrature |
| open scoped Classical BigOperators NNReal |
|
|
| def kernelQuadraturePicard {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (r T h ψ : ℝ) (n m N : ℕ) (e : ProbabilityNode T h n) |
| (w : ProbabilityNode T h n → ℝ) (p : Point d × Point d) : Point d := |
| ∑s,w s • kernelActionPicard F x r T h ψ n m N s e p |
|
|
| theorem kernelQuadraturePicard_numerical_rms (n : ℕ) (hn : 0<n) : |
| ∃Ap Ah Lp Lh : ℝ≥0,∃C J : ℝ,0≤C ∧ 1≤J ∧ ∃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 → |
| Ap*probabilityMeanLipschitz lam r≤1/2 → Ah*probabilityMeanLipschitz lam r≤1/2 → |
| Lp*probabilityMeanLipschitz lam r≤1/2 → Lh*probabilityMeanLipschitz lam r≤1/2 → |
| ∀N : ℕ,∀R : ℝ,0<R → R^2≤1-T^2 → ∀m : ℕ,1 ≤ m → ∀ψ : ℝ,0<ψ → ψ*m≤1 → |
| ∀w : ProbabilityNode T h n → ℝ, |
| let err := fun p : Point d × Point d => |
| kernelQuadraturePicard F x r T h ψ n m N (probabilityEndpoint hT0 hT1 hh n) w p- |
| ∑s,w s • (terminalJacobian hF x hr hl hT0.le hT1 |
| (probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2 |
| Integrable (fun p => ‖err p‖^2) ((interpolationLaw F x r T).prod (stdGaussian (Point d))) ∧ |
| (∫p,‖err p‖^2 ∂((interpolationLaw F x r T).prod (stdGaussian (Point d))))≤ |
| (∑s,|w s|)*(∑s,|w s| * |
| kernelActionRmsBudget F x lam Ap Ah Lp r R (probabilityNodeTime T h n s) h ψ C J k n m N) := by |
| obtain ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,hA⟩ := kernelActionPicard_terminal_rms n hn |
| refine ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,?_⟩ |
| intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hqp hqh hqlp hqlh N R hR hRT m hm1 ψ hψ hψm w |
| dsimp only |
| let ep := probabilityEndpoint hT0 hT1 hh n |
| let f := fun (s : ProbabilityNode T h n) (p : Point d × Point d) => |
| kernelActionPicard F x r T h ψ n m N s ep p- |
| (terminalJacobian hF x hr hl hT0.le hT1 |
| (probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2 |
| have hs (s : ProbabilityNode T h n) := |
| hA hF x hr hlam hl hd hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s R hR hRT m hm1 ψ hψ hψm |
| have hfc (s : ProbabilityNode T h n) : Continuous (f s) := by |
| have hK : Continuous (fun p : Point d × Point d => |
| (terminalJacobian hF x hr hl hT0.le hT1 |
| (probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2) := |
| (((terminalJacobian_smooth hF x hr hl hT0.le hT1).continuous.comp |
| (continuous_const.prodMk continuous_fst)).sub continuous_const).clm_apply continuous_snd |
| exact (kernelActionPicard_continuous hF x hr.le hl hT0 hT1 hh ψ n hn m N s ep).sub hK |
| have he (p : Point d × Point d) : |
| kernelQuadraturePicard F x r T h ψ n m N ep w p- |
| ∑s,w s • (terminalJacobian hF x hr hl hT0.le hT1 |
| (probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2= |
| ∑s,w s • f s p := by |
| simp only [kernelQuadraturePicard,f,smul_sub,Finset.sum_sub_distrib] |
| simp_rw [show probabilityEndpoint hT0 hT1 hh n=ep from rfl,he] |
| exact weighted_sum_sq_varying w f |
| (fun s => kernelActionRmsBudget F x lam Ap Ah Lp r R (probabilityNodeTime T h n s) h ψ C J k n m N) |
| (fun s => (hfc s).aestronglyMeasurable) (fun s => (hs s).1) (fun s => (hs s).2) |
| end LogConcaveSampling |
|
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| end |
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| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Quadrature |
| open scoped Classical BigOperators NNReal |
|
|
| theorem kernelQuadraturePicard_lipschitz (n : ℕ) (hn : 0<n) : |
| ∃A B : ℝ≥0,∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, |
| Primitive F lam → ∀(x : Point d) {r T h ψ : ℝ},0≤r → |
| (lam:ℝ)*r^2≤1/2 → 0<T → T<1 → 0<h → 0<ψ → |
| ∀m : ℕ,ψ*m≤1 → |
| A*probabilityMeanLipschitz lam r≤1/2 → |
| B*probabilityMeanLipschitz lam r≤1/2 → |
| ∀N : ℕ,∀e : ProbabilityNode T h n,∀w : ProbabilityNode T h n → ℝ, |
| LipschitzWith ((∑s,‖w s‖₊)* |
| ((∑i : Fin (m+1),‖derivativeWeight (angleNodes m) i/ψ‖₊)* |
| (16*(A*probabilityMeanLipschitz lam r)))) |
| (kernelQuadraturePicard F x r T h ψ n m N e w) := by |
| obtain ⟨A,B,hAB⟩ := kernelActionPicard_lipschitz n hn |
| refine ⟨A,B,?_⟩ |
| intro d F lam hF x r T h ψ hr hl hT0 hT1 hh hψ m hψm hq hq' N e w |
| exact finite_weighted_lipschitz w |
| (fun s => kernelActionPicard F x r T h ψ n m N s e) |
| (fun s => hAB hF x hr hl hT0 hT1 hh hψ m hψm hq hq' N s e) |
| end LogConcaveSampling |
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|
| end |
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| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.FinitePicard |
| open MeasureTheory |
| open scoped BigOperators NNReal RealInnerProductSpace |
|
|
| variable {Ω E I : Type*} [MeasurableSpace Ω] {ν : Measure Ω} [IsFiniteMeasure ν] |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fintype I] |
|
|
| lemma iterateLp_ae (w : I → I → ℝ) (φ : I → E → E) {K : ℝ≥0} |
| (hφ : ∀i,LipschitzWith K (φ i)) (a : I → Lp E 2 ν) (n : ℕ) : |
| ∀ᵐω ∂ν,∀i, ((stepLp w φ hφ a)^[n] a i) ω= |
| nodes w φ (fun ω j => a j ω) n ω i := by |
| induction n with |
| | zero => exact Filter.Eventually.of_forall (fun _ _ => rfl) |
| | succ n ih => |
| have hs : ∀ᵐω ∂ν,∀i, |
| (stepLp w φ hφ a ((stepLp w φ hφ a)^[n] a) i) ω= |
| step w φ (fun j => a j ω) (fun j => (((stepLp w φ hφ a)^[n] a) j) ω) i := by |
| rw [ae_all_iff] |
| exact stepLp_ae w φ hφ a _ |
| filter_upwards [hs,ih] with ω hs hi |
| intro i |
| rw [Function.iterate_succ_apply',hs i] |
| simp only [nodes,hi] |
|
|
| theorem actual_nodes_rms (w : I → I → ℝ) (φ : I → E → E) {A K : ℝ≥0} |
| (hφ : ∀i,LipschitzWith K (φ i)) (a e : I → Lp E 2 ν) |
| (hw : ∀i,∑j,|w i j|≤A) (hq : (A*K:ℝ≥0)≤1/2) |
| {D : ℝ} (hD : 0≤D) |
| (he : ∀i,(∫ω,‖(stepLp w φ hφ a e i) ω-e i ω‖^2 ∂ν)≤D^2) |
| (n : ℕ) (i : I) : |
| (∫ω,‖nodes w φ (fun ω j => a j ω) n ω i-e i ω‖^2 ∂ν)≤ |
| (2*D+(A*K:ℝ≥0)^n*dist a e)^2 := by |
| have hh := rms_picard_defect w φ hφ a e hw hq hD |
| (fun i => nodewise_rms_le hD (he i)) n |
| have hj := (dist_le_pi_dist ((stepLp w φ hφ a)^[n] a) e i).trans hh |
| have hn : 0≤2*D+(A*K:ℝ≥0)^n*dist a e := by positivity |
| have heq : (∫ω,‖nodes w φ (fun ω j => a j ω) n ω i-e i ω‖^2 ∂ν)= |
| dist (((stepLp w φ hφ a)^[n] a) i) (e i)^2 := by |
| rw [dist_eq_norm,lp_norm_sq_integral] |
| apply integral_congr_ae |
| filter_upwards [iterateLp_ae w φ hφ a n, |
| Lp.coeFn_sub (((stepLp w φ hφ a)^[n] a) i) (e i)] with ω h1 h2 |
| simp only [h2,Pi.sub_apply,h1 i] |
| rw [heq] |
| exact pow_le_pow_left₀ dist_nonneg hj 2 |
| end LogConcaveSampling.FinitePicard |
|
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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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